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Theorem suctrALTcfVD 28072
Description: The following User's Proof is a Virtual Deduction proof ( see: wvd1 27710) using conjunction-form virtual hypothesis collections. The conjunction-form version of completeusersproof.cmd. It allows the User to avoid superflous virtual hypotheses. This proof was completed automatically by a tools program which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. suctrALTcf 28071 is suctrALTcfVD 28072 without virtual deductions and was derived automatically from suctrALTcfVD 28072. The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  (. Tr  A  ->.  Tr  A ).
2::  |-  (..........  ( z  e.  y  /\  y  e.  suc  A )  ->.  ( z  e.  y  /\  y  e.  suc  A ) ).
3:2:  |-  (..........  ( z  e.  y  /\  y  e.  suc  A )  ->.  z  e.  y ).
4::  |-  (.................................... .......  y  e.  A  ->.  y  e.  A ).
5:1,3,4:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ,  y  e.  A ).  ->.  z  e.  A ).
6::  |-  A  C_  suc  A
7:5,6:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ,  y  e.  A ).  ->.  z  e.  suc  A ).
8:7:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ).  ->.  ( y  e.  A  ->  z  e.  suc  A ) ).
9::  |-  (.................................... ......  y  =  A  ->.  y  =  A ).
10:3,9:  |-  (.........  (. ( z  e.  y  /\  y  e.  suc  A ) ,  y  =  A ).  ->.  z  e.  A ).
11:10,6:  |-  (.........  (. ( z  e.  y  /\  y  e.  suc  A ) ,  y  =  A ).  ->.  z  e.  suc  A ).
12:11:  |-  (...........  ( z  e.  y  /\  y  e.  suc  A )  ->.  ( y  =  A  ->  z  e.  suc  A ) ).
13:2:  |-  (...........  ( z  e.  y  /\  y  e.  suc  A )  ->.  y  e.  suc  A ).
14:13:  |-  (...........  ( z  e.  y  /\  y  e.  suc  A )  ->.  ( y  e.  A  \/  y  =  A ) ).
15:8,12,14:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ).  ->.  z  e.  suc  A ).
16:15:  |-  (. Tr  A  ->.  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) ).
17:16:  |-  (. Tr  A  ->.  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) ).
18:17:  |-  (. Tr  A  ->.  Tr  suc  A ).
qed:18:  |-  ( Tr  A  ->  Tr  suc  A )
Assertion
Ref Expression
suctrALTcfVD  |-  ( Tr  A  ->  Tr  suc  A
)

Proof of Theorem suctrALTcfVD
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 4469 . . . . . . . 8  |-  A  C_  suc  A
2 idn1 27715 . . . . . . . . 9  |-  (. Tr  A 
->.  Tr  A ).
3 idn1 27715 . . . . . . . . . 10  |-  (. (
z  e.  y  /\  y  e.  suc  A )  ->.  ( z  e.  y  /\  y  e.  suc  A ) ).
4 simpl 443 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
53, 4el1 27773 . . . . . . . . 9  |-  (. (
z  e.  y  /\  y  e.  suc  A )  ->.  z  e.  y ).
6 idn1 27715 . . . . . . . . 9  |-  (. y  e.  A  ->.  y  e.  A ).
7 trel 4120 . . . . . . . . . 10  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
873impib 1149 . . . . . . . . 9  |-  ( ( Tr  A  /\  z  e.  y  /\  y  e.  A )  ->  z  e.  A )
92, 5, 6, 8el123 27912 . . . . . . . 8  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A ) ,. y  e.  A ).  ->.  z  e.  A ).
10 ssel2 3175 . . . . . . . 8  |-  ( ( A  C_  suc  A  /\  z  e.  A )  ->  z  e.  suc  A
)
111, 9, 10el0321old 27869 . . . . . . 7  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A ) ,. y  e.  A ).  ->.  z  e.  suc  A ).
1211int3 27757 . . . . . 6  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A ) ).  ->.  ( y  e.  A  ->  z  e.  suc  A
) ).
13 idn1 27715 . . . . . . . . 9  |-  (. y  =  A  ->.  y  =  A ).
14 eleq2 2344 . . . . . . . . . 10  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
1514biimpac 472 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  =  A )  ->  z  e.  A )
165, 13, 15el12 27874 . . . . . . . 8  |-  (. (. ( z  e.  y  /\  y  e.  suc  A ) ,. y  =  A ).  ->.  z  e.  A ).
171, 16, 10el021old 27847 . . . . . . 7  |-  (. (. ( z  e.  y  /\  y  e.  suc  A ) ,. y  =  A ).  ->.  z  e.  suc  A ).
1817int2 27751 . . . . . 6  |-  (. (
z  e.  y  /\  y  e.  suc  A )  ->.  ( y  =  A  ->  z  e.  suc  A ) ).
19 simpr 447 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
203, 19el1 27773 . . . . . . 7  |-  (. (
z  e.  y  /\  y  e.  suc  A )  ->.  y  e.  suc  A ).
21 elsuci 4458 . . . . . . 7  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
2220, 21el1 27773 . . . . . 6  |-  (. (
z  e.  y  /\  y  e.  suc  A )  ->.  ( y  e.  A  \/  y  =  A
) ).
23 jao 498 . . . . . . 7  |-  ( ( y  e.  A  -> 
z  e.  suc  A
)  ->  ( (
y  =  A  -> 
z  e.  suc  A
)  ->  ( (
y  e.  A  \/  y  =  A )  ->  z  e.  suc  A
) ) )
24233imp 1145 . . . . . 6  |-  ( ( ( y  e.  A  ->  z  e.  suc  A
)  /\  ( y  =  A  ->  z  e. 
suc  A )  /\  ( y  e.  A  \/  y  =  A
) )  ->  z  e.  suc  A )
2512, 18, 22, 24el2122old 27871 . . . . 5  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A ) ).  ->.  z  e.  suc  A ).
2625int2 27751 . . . 4  |-  (. Tr  A 
->.  ( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) ).
2726gen12 27763 . . 3  |-  (. Tr  A 
->.  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) ).
28 dftr2 4115 . . . 4  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2928biimpri 197 . . 3  |-  ( A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A )  ->  Tr  suc  A
)
3027, 29el1 27773 . 2  |-  (. Tr  A 
->.  Tr  suc  A ).
3130in1 27712 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684    C_ wss 3152   Tr wtr 4113   suc csuc 4394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-sn 3646  df-uni 3828  df-tr 4114  df-suc 4398  df-vd1 27711  df-vhc2 27723  df-vhc3 27731
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