Type  Label  Description 
Statement 

Theorem  pwtrrOLD 28601 
A set is transitive if its power set is. The proof of this theorem was
automatically generated from pwtrrVD 28600 using a tools command file,
translateMWO.cmd , by translating the proof into its nonvirtual
deduction form and minimizing it. (Contributed by Alan Sare,
25Aug2011.) (Moved into main set.mm as pwtr 4226
and may be deleted by
mathbox owner, AS. NM 15Jun2014.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  snssiALTVD 28602 
Virtual deduction proof of snssiALT 28603. (Contributed by Alan Sare,
11Sep2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  snssiALT 28603 
If a class is an element of another class, then its singleton is a
subclass of that other class. Alternate proof of snssi 3759. This
theorem was automatically generated from snssiALTVD 28602 using a
translation program. (Contributed by Alan Sare, 11Sep2011.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  snsslVD 28604 
Virtual deduction proof of snssl 28605. (Contributed by Alan Sare,
25Aug2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  snssl 28605 
If a singleton is a subclass of another class, then the singleton's
element is an element of that other class. This theorem is the
righttoleft implication of the biconditional snss 3748.
The proof of
this theorem was automatically generated from snsslVD 28604 using a tools
command file, translateMWO.cmd , by translating the proof into its
nonvirtual deduction form and minimizing it. (Contributed by Alan
Sare, 25Aug2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  snelpwrVD 28606 
Virtual deduction proof of snelpwi 4220. (Contributed by Alan Sare,
25Aug2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  snelpwrOLD 28607 
If a class is contained in another class, then its singleton is
contained in the power class of that other class. This theorem is the
lefttoright implication of the biconditional snelpw 4221. Unlike
snelpw 4221, may be a proper class. The proof of this theorem was
automatically generated from snelpwrVD 28606 using a tools command file,
translateMWO.cmd , by translating the proof into its nonvirtual
deduction form and minimizing it. (Moved to snelpwi 4220 in main set.mm
and may be deleted by mathbox owner, AS. NM 10Sep2013.)
(Contributed by Alan Sare, 25Aug2011.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  unipwrVD 28608 
Virtual deduction proof of unipwr 28609. (Contributed by Alan Sare,
25Aug2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  unipwr 28609 
A class is a subclass of the union of its power class. This theorem is
the righttoleft subclass lemma of unipw 4224. The proof of this theorem
was automatically generated from unipwrVD 28608 using a tools command file ,
translateMWO.cmd , by translating the proof into its nonvirtual
deduction form and minimizing it. (Contributed by Alan Sare,
25Aug2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  sstrALT2VD 28610 
Virtual deduction proof of sstrALT2 28611. (Contributed by Alan Sare,
11Sep2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  sstrALT2 28611 
Virtual deduction proof of sstr 3187, transitivity of subclasses, Theorem
6 of [Suppes] p. 23. This theorem was
automatically generated from
sstrALT2VD 28610 using the command file translate_{without}_overwriting.cmd .
It was not minimized because the automated minimization excluding
duplicates generates a minimized proof which, although not directly
containing any duplicates, indirectly contains a duplicate. That is,
the trace back of the minimized proof contains a duplicate. This is
undesirable because some step(s) of the minimized proof use the proven
theorem. (Contributed by Alan Sare, 11Sep2011.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  suctrALT2VD 28612 
Virtual deduction proof of suctrALT2 28613. (Contributed by Alan Sare,
11Sep2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  suctrALT2 28613 
Virtual deduction proof of suctr 4475. The sucessor of a transitive class
is transitive. This proof was generated automatically from the virtual
deduction proof suctrALT2VD 28612 using the tools command file
translate_{without}_overwriting_{minimize}_excluding_{duplicates}.cmd .
(Contributed by Alan Sare, 11Sep2011.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  elex2VD 28614* 
Virtual deduction proof of elex2 2800. (Contributed by Alan Sare,
25Sep2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  elex22VD 28615* 
Virtual deduction proof of elex22 2799. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  eqsbc3rVD 28616* 
Virtual deduction proof of eqsbc3r 3048. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  zfregs2VD 28617* 
Virtual deduction proof of zfregs2 7415. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  tpid3gVD 28618 
Virtual deduction proof of tpid3g 3741. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  en3lplem1VD 28619* 
Virtual deduction proof of en3lplem1 7416. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  en3lplem2VD 28620* 
Virtual deduction proof of en3lplem2 7417. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  en3lpVD 28621 
Virtual deduction proof of en3lp 7418. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



18.25.5 Theorems proved using virtual deduction
with mmj2 assistance


Theorem  simplbi2VD 28622 
Virtual deduction proof of simplbi2 608. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
h1:: 
 3:1,?: e0_ 28547 
 qed:3,?: e0_ 28547 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3ornot23VD 28623 
Virtual deduction proof of 3ornot23 28270. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1::
 2:: 
 3:1,?: e1_ 28399 
 4:1,?: e1_ 28399 
 5:3,4,?: e11 28460 
 6:2,?: e2 28403 
 7:5,6,?: e12 28499 
 8:7: 
 qed:8: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  orbi1rVD 28624 
Virtual deduction proof of orbi1r 28271. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:2,?: e2 28403 
 4:1,3,?: e12 28499 
 5:4,?: e2 28403 
 6:5: 
 7:: 
 8:7,?: e2 28403 
 9:1,8,?: e12 28499 
 10:9,?: e2 28403 
 11:10: 
 12:6,11,?: e11 28460 
 qed:12: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  bitr3VD 28625 
Virtual deduction proof of bitr3 28272. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1_ 28399 
 3:: 
 4:3,?: e2 28403 
 5:2,4,?: e12 28499 
 6:5: 
 qed:6: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3orbi123VD 28626 
Virtual deduction proof of 3orbi123 28273. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1_ 28399 
 3:1,?: e1_ 28399 
 4:1,?: e1_ 28399 
 5:2,3,?: e11 28460 
 6:5,4,?: e11 28460 
 7:?: 
 8:6,7,?: e10 28467 
 9:?: 
 10:8,9,?: e10 28467 
 qed:10: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sbc3orgVD 28627 
Virtual deduction proof of sbc3org 28295. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1_ 28399 
 3:: 
 32:3: 
 33:1,32,?: e10 28467 
 4:1,33,?: e11 28460 
 5:2,4,?: e11 28460 
 6:1,?: e1_ 28399 
 7:6,?: e1_ 28399 
 8:5,7,?: e11 28460 
 9:?: 
 10:8,9,?: e10 28467 
 qed:10: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  19.21a3con13vVD 28628* 
Virtual deduction proof of alrim3con13v 28296. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:2,?: e2 28403 
 4:2,?: e2 28403 
 5:2,?: e2 28403 
 6:1,4,?: e12 28499 
 7:3,?: e2 28403 
 8:5,?: e2 28403 
 9:7,6,8,?: e222 28408 
 10:9,?: e2 28403 
 11:10:in2 
 qed:11:in1 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  exbirVD 28629 
Virtual deduction proof of exbir 1355. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:: 
 5:1,2,?: e12 28499 
 6:3,5,?: e32 28533 
 7:6: 
 8:7: 
 9:8,?: e1_ 28399 
 qed:9: 

(Contributed by Alan Sare, 13Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  exbiriVD 28630 
Virtual deduction proof of exbiri 605. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
h1:: 
 2:: 
 3:: 
 4:: 
 5:2,1,?: e10 28467 
 6:3,5,?: e21 28505 
 7:4,6,?: e32 28533 
 8:7: 
 9:8: 
 qed:9: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  rspsbc2VD 28631* 
Virtual deduction proof of rspsbc2 28297. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:: 
 4:1,3,?: e13 28523 
 5:1,4,?: e13 28523 
 6:2,5,?: e23 28530 
 7:6: 
 8:7: 
 qed:8: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3impexpVD 28632 
Virtual deduction proof of 3impexp 1356. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:1,2,?: e10 28467 
 4:3,?: e1_ 28399 
 5:4,?: e1_ 28399 
 6:5: 
 7:: 
 8:7,?: e1_ 28399 
 9:8,?: e1_ 28399 
 10:2,9,?: e01 28463 
 11:10: 
 qed:6,11,?: e00 28543 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3impexpbicomVD 28633 
Virtual deduction proof of 3impexpbicom 1357. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:1,2,?: e10 28467 
 4:3,?: e1_ 28399 
 5:4: 
 6:: 
 7:6,?: e1_ 28399 
 8:7,2,?: e10 28467 
 9:8: 
 qed:5,9,?: e00 28543 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3impexpbicomiVD 28634 
Virtual deduction proof of 3impexpbicomi 1358. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sbcoreleleqVD 28635* 
Virtual deduction proof of sbcoreleleq 28298. The following user's proof
is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1_ 28399 
 3:1,?: e1_ 28399 
 4:1,?: e1_ 28399 
 5:2,3,4,?: e111 28446 
 6:1,?: e1_ 28399 
 7:5,6: e11 28460 
 qed:7: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  hbra2VD 28636* 
Virtual deduction proof of nfra2 2597. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:1,2,?: e00 28543 
 4:2: 
 5:4,?: e0_ 28547 
 qed:3,5,?: e00 28543 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  tratrbVD 28637* 
Virtual deduction proof of tratrb 28299. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1_ 28399 
 3:1,?: e1_ 28399 
 4:1,?: e1_ 28399 
 5:: 
 6:5,?: e2 28403 
 7:5,?: e2 28403 
 8:2,7,4,?: e121 28428 
 9:2,6,8,?: e122 28425 
 10:: 
 11:6,7,10,?: e223 28407 
 12:11: 
 13:: 
 14:12,13,?: e20 28502 
 15:: 
 16:7,15,?: e23 28530 
 17:6,16,?: e23 28530 
 18:17: 
 19:: 
 20:18,19,?: e20 28502 
 21:3,?: e1_ 28399 
 22:21,9,4,?: e121 28428 
 23:22,?: e2 28403 
 24:4,23,?: e12 28499 
 25:14,20,24,?: e222 28408 
 26:25: 
 27:: 
 28:27,?: e0_ 28547 
 29:28,26: 
 30:: 
 31:30,?: e0_ 28547 
 32:31,29: 
 33:32,?: e1_ 28399 
 qed:33: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3ax5VD 28638 
Virtual deduction proof of 3ax5 28300. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1_ 28399 
 3:: 
 4:2,3,?: e10 28467 
 qed:4: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  syl5impVD 28639 
Virtual deduction proof of syl5imp 28274. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1_ 28399 
 3:: 
 4:3,2,?: e21 28505 
 5:4,?: e2 28403 
 6:5: 
 qed:6: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  idiVD 28640 
Virtual deduction proof of idi 2. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  ancomsimpVD 28641 
Closed form of ancoms 439. The following user's proof is completed by
invoking mmj2's unify command and using mmj2's StepSelector to pick all
remaining steps of the Metamath proof.
(Contributed by Alan Sare, 25Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  ssralv2VD 28642* 
Quantification restricted to a subclass for two quantifiers. ssralv 3237
for two quantifiers. The following User's Proof is a Virtual Deduction
proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. ssralv2 28294 is ssralv2VD 28642 without
virtual deductions and was automatically derived from ssralv2VD 28642.
1:: 
 2:: 
 3:1: 
 4:3,2: 
 5:4: 
 6:5: 
 7:: 
 8:7,6: 
 9:1: 
 10:9,8: 
 11:10: 
 12:: 
 13:: 
 14:12,13,11: 
 15:14: 
 16:15: 
 qed:16: 

(Contributed by Alan Sare, 10Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  ordelordALTVD 28643 
An element of an ordinal class is ordinal. Proposition 7.6 of
[TakeutiZaring] p. 36. This is an alternate proof of ordelord 4414 using
the Axiom of Regularity indirectly through dford2 7321. dford2 is a
weaker definition of ordinal number. Given the Axiom of Regularity, it
need not be assumed that because this is inferred by the
Axiom of Regularity. The following User's Proof is a Virtual Deduction
proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. ordelordALT 28301 is ordelordALTVD 28643
without virtual deductions and was automatically derived from
ordelordALTVD 28643 using the tools program
translate..without..overwriting.cmd and Metamath's minimize command.
1:: 
 2:1: 
 3:1: 
 4:2: 
 5:2: 
 6:4,3: 
 7:6,6,5: 
 8:: 
 9:8: 
 10:9: 
 11:10: 
 12:11: 
 13:12: 
 14:13: 
 15:14,5: 
 16:4,15,3: 
 17:16,7: 
 qed:17: 

(Contributed by Alan Sare, 12Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  equncomVD 28644 
If a class equals the union of two other classes, then it equals the
union of those two classes commuted. The following User's Proof is a
Virtual Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. equncom 3320 is equncomVD 28644 without
virtual deductions and was automatically derived from equncomVD 28644.
1:: 
 2:: 
 3:1,2: 
 4:3: 
 5:: 
 6:5,2: 
 7:6: 
 8:4,7: 

(Contributed by Alan Sare, 17Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  equncomiVD 28645 
Inference form of equncom 3320. The following User's Proof is a
Virtual Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. equncomi 3321 is equncomiVD 28645 without
virtual deductions and was automatically derived from equncomiVD 28645.
(Contributed by Alan Sare, 18Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sucidALTVD 28646 
A set belongs to its successor. Alternate proof of sucid 4471.
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program
completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. sucidALT 28647 is sucidALTVD 28646
without virtual deductions and was automatically derived from
sucidALTVD 28646. This proof illustrates that
completeusersproof.cmd will generate a Metamath proof from any
User's Proof which is "conventional" in the sense that no step
is a virtual deduction, provided that all necessary unification
theorems and transformation deductions are in set.mm.
completeusersproof.cmd automatically converts such a
conventional proof into a Virtual Deduction proof for which each
step happens to be a 0virtual hypothesis virtual deduction.
The user does not need to search for reference theorem labels or
deduction labels nor does he(she) need to use theorems and
deductions which unify with reference theorems and deductions in
set.mm. All that is necessary is that each theorem or deduction
of the User's Proof unifies with some reference theorem or
deduction in set.mm or is a semantic variation of some theorem
or deduction which unifies with some reference theorem or
deduction in set.mm. The definition of "semantic variation" has
not been precisely defined. If it is obvious that a theorem or
deduction has the same meaning as another theorem or deduction,
then it is a semantic variation of the latter theorem or
deduction. For example, step 4 of the User's Proof is a
semantic variation of the definition (axiom)
, which unifies with dfsuc 4398, a
reference definition (axiom) in set.mm. Also, a theorem or
deduction is said to be a semantic variation of a another
theorem or deduction if it is obvious upon cursory inspection
that it has the same meaning as a weaker form of the latter
theorem or deduction. For example, the deduction
infers is a
semantic variation of the theorem
, which unifies with
the set.mm reference definition (axiom) dford2 7321.
h1:: 
 2:1: 
 3:2: 
 4:: 
 qed:3,4: 

(Contributed by Alan Sare, 18Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sucidALT 28647 
A set belongs to its successor. This proof was automatically derived
from sucidALTVD 28646 using translate_{without}_overwriting.cmd and
minimizing. (Contributed by Alan Sare, 18Feb2012.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  sucidVD 28648 
A set belongs to its successor. The following User's Proof is a
Virtual Deduction proof completed automatically by the tools
program completeusersproof.cmd, which invokes Mel O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant. sucid 4471 is
sucidVD 28648 without virtual deductions and was automatically
derived from sucidVD 28648.
h1:: 
 2:1: 
 3:2: 
 4:: 
 qed:3,4: 

(Contributed by Alan Sare, 18Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  imbi12VD 28649 
Implication form of imbi12i 316. The following User's Proof is a Virtual
Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi12 28282
is imbi12VD 28649 without virtual deductions and was automatically derived
from imbi12VD 28649.
1:: 
 2:: 
 3:: 
 4:1,3: 
 5:2,4: 
 6:5: 
 7:: 
 8:1,7: 
 9:2,8: 
 10:9: 
 11:6,10: 
 12:11: 
 qed:12: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  imbi13VD 28650 
Join three logical equivalences to form equivalence of implications. The
following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi13 28283
is imbi13VD 28650 without virtual deductions and was automatically derived
from imbi13VD 28650.
1:: 
 2:: 
 3:: 
 4:2,3: 
 5:1,4: 
 6:5: 
 7:6: 
 qed:7: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sbcim2gVD 28651 
Distribution of class substitution over a leftnested implication.
Similar
to sbcimg 3032. The following User's Proof is a Virtual Deduction proof
completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcim2g 28302
is sbcim2gVD 28651 without virtual deductions and was automatically derived
from sbcim2gVD 28651.
1:: 
 2:: 
 3:1,2: 
 4:1: 
 5:3,4: 
 6:5: 
 7:: 
 8:4,7: 
 9:1: 
 10:8,9: 
 11:10: 
 12:6,11: 
 qed:12: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sbcbiVD 28652 
Implication form of sbcbiiOLD 3047. The following User's Proof is a
Virtual
Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcbi 28303
is sbcbiVD 28652 without virtual deductions and was automatically derived
from sbcbiVD 28652.
1:: 
 2:: 
 3:1,2: 
 4:1,3: 
 5:4: 
 qed:5: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  trsbcVD 28653* 
Formulabuilding inference rule for class substitution, substituting a
class
variable for the set variable of the transitivity predicate. The
following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trsbc 28304
is trsbcVD 28653 without virtual deductions and was automatically derived
from trsbcVD 28653.
1:: 
 2:1: 
 3:1: 
 4:1: 
 5:1,2,3,4: 
 6:1: 
 7:5,6: 
 8:: 
 9:7,8: 
 10:: 
 11:10: 
 12:1,11: 
 13:9,12: 
 14:13: 
 15:14: 
 16:1: 
 17:15,16: 
 18:17: 
 19:18: 
 20:1: 
 21:19,20: 
 22:: 
 23:21,22: 
 24:: 
 25:24: 
 26:1,25: 
 27:23,26: 
 qed:27: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  truniALTVD 28654* 
The union of a class of transitive sets is transitive. The following
User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. truniALT 28305
is truniALTVD 28654 without virtual deductions and was automatically derived
from truniALTVD 28654.
1:: 
 2:: 
 3:2: 
 4:2: 
 5:4: 
 6:: 
 7:6: 
 8:6: 
 9:1,8: 
 10:8,9: 
 11:3,7,10: 
 12:11,8: 
 13:12: 
 14:13: 
 15:14: 
 16:5,15: 
 17:16: 
 18:17: 
 19:18: 
 qed:19: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  ee33VD 28655 
Nonvirtual deduction form of e33 28509. The following User's Proof is a
Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ee33 28284
is ee33VD 28655 without virtual deductions and was automatically derived
from ee33VD 28655.
h1:: 
 h2:: 
 h3:: 
 4:1,3: 
 5:4: 
 6:2,5: 
 7:6: 
 8:7: 
 qed:8: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  trintALTVD 28656* 
The intersection of a class of transitive sets is transitive. Virtual
deduction proof of trintALT 28657.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trintALT 28657
is trintALTVD 28656 without virtual deductions and was automatically derived
from trintALTVD 28656.
1:: 
 2:: 
 3:2: 
 4:2: 
 5:4: 
 6:5: 
 7:: 
 8:7,6: 
 9:7,1: 
 10:7,9: 
 11:10,3,8: 
 12:11: 
 13:12: 
 14:13: 
 15:3,14: 
 16:15: 
 17:16: 
 18:17: 
 qed:18: 

(Contributed by Alan Sare, 17Apr2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  trintALT 28657* 
The intersection of a class of transitive sets is transitive. Exercise
5(b) of [Enderton] p. 73. trintALT 28657 is an alternative proof of
trint 4128. trintALT 28657 is trintALTVD 28656 without virtual deductions and was
automatically derived from trintALTVD 28656 using the tools program
translate..without..overwriting.cmd and Metamath's minimize command.
(Contributed by Alan Sare, 17Apr2012.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  undif3VD 28658 
The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual
deduction proof of undif3 3429.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. undif3 3429
is undif3VD 28658 without virtual deductions and was automatically derived
from undif3VD 28658.
1:: 
 2:: 
 3:2: 
 4:1,3: 
 5:: 
 6:5: 
 7:5: 
 8:6,7: 
 9:8: 
 10:: 
 11:10: 
 12:10: 
 13:11: 
 14:12: 
 15:13,14: 
 16:15: 
 17:9,16: 
 18:: 
 19:18: 
 20:18: 
 21:18: 
 22:21: 
 23:: 
 24:23: 
 25:24: 
 26:25: 
 27:10: 
 28:27: 
 29:: 
 30:29: 
 31:30: 
 32:31: 
 33:22,26: 
 34:28,32: 
 35:33,34: 
 36:: 
 37:36,35: 
 38:17,37: 
 39:: 
 40:39: 
 41:: 
 42:40,41: 
 43:: 
 44:43,42: 
 45:: 
 46:45,44: 
 47:4,38: 
 48:46,47: 
 49:48: 
 qed:49: 

(Contributed by Alan Sare, 17Apr2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sbcssVD 28659 
Virtual deduction proof of sbcss 3564.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcss 3564
is sbcssVD 28659 without virtual deductions and was automatically derived
from sbcssVD 28659.
1:: 
 2:1: 
 3:1: 
 4:2,3: 
 5:1: 
 6:4,5: 
 7:6: 
 8:7: 
 9:1: 
 10:8,9: 
 11:: 
 110:11: 
 12:1,110: 
 13:10,12: 
 14:: 
 15:13,14: 
 qed:15: 

(Contributed by Alan Sare, 22Jul2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  csbingVD 28660 
Virtual deduction proof of csbing 3376.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbing 3376
is csbingVD 28660 without virtual deductions and was automatically derived
from csbingVD 28660.
1:: 
 2:: 
 20:2: 
 30:1,20: 
 3:1,30: 
 4:1: 
 5:3,4: 
 6:1: 
 7:1: 
 8:6,7: 
 9:1: 
 10:9,8: 
 11:10: 
 12:11: 
 13:5,12: 
 14:: 
 15:13,14: 
 qed:15: 

(Contributed by Alan Sare, 22Jul2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  onfrALTlem5VD 28661* 
Virtual deduction proof of onfrALTlem5 28307.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem5 28307 is onfrALTlem5VD 28661 without virtual deductions and was
automatically derived
from onfrALTlem5VD 28661.
1:: 
 2:1: 
 3:2: 
 4:3: 
 5:: 
 6:4,5: 
 7:2: 
 8:: 
 9:8: 
 10:2,9: 
 11:7,10: 
 12:6,11: 
 13:2: 
 14:12,13: 
 15:2: 
 16:15,14: 
 17:2: 
 18:2: 
 19:2: 
 20:18,19: 
 21:17,20: 
 22:2: 
 23:2: 
 24:21,23: 
 25:22,24: 
 26:2: 
 27:25,26: 
 28:2: 
 29:27,28: 
 30:29: 
 31:30: 
 32:: 
 33:31,32: 
 34:2: 
 35:33,34: 
 36:: 
 37:36: 
 38:2,37: 
 39:35,38: 
 40:16,39: 
 