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Theorem onfrALTVD 28983
Description: Virtual deduction proof of onfrALT 28613. 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. onfrALT 28613 is onfrALTVD 28983 without virtual deductions and was automatically derived from onfrALTVD 28983.
1::  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  E. y  e.  a ( a  i^i  y )  =  (/) ).
2::  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->.  E. y  e.  a ( a  i^i  y )  =  (/) ).
3:1:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( -.  ( a  i^i  x )  =  (/)  ->  E. y  e.  a ( a  i^i  y )  =  (/) ) ).
4:2:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( ( a  i^i  x )  =  (/)  ->  E. y  e.  a ( a  i^i  y )  =  (/) ) ).
5::  |-  ( ( a  i^i  x )  =  (/)  \/  -.  ( a  i^i  x )  =  (/) )
6:5,4,3:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  E. y  e.  a ( a  i^i  y )  =  (/) ).
7:6:  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  ( x  e.  a  ->  E. y  e.  a ( a  i^i  y )  =  (/) ) ).
8:7:  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  A. x ( x  e.  a  ->  E. y  e.  a ( a  i^i  y )  =  (/) ) ).
9:8:  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  ( E. x x  e.  a  ->  E. y  e.  a ( a  i^i  y )  =  (/) ) ).
10::  |-  ( a  =/=  (/)  <->  E. x x  e.  a )
11:9,10:  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  ( a  =/=  (/)  ->  E. y  e.  a ( a  i^i  y )  =  (/) ) ).
12::  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
13:12:  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  a  =/=  (/) ).
14:13,11:  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  E. y  e.  a ( a  i^i  y )  =  (/) ).
15:14:  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
16:15:  |-  A. a ( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a ( a  i^i  y )  =  (/) )
qed:16:  |-  _E  Fr  On
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTVD  |-  _E  Fr  On

Proof of Theorem onfrALTVD
Dummy variables  x  a  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 28641 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
2 simpr 447 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  =/=  (/) )
31, 2e1_ 28704 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  a  =/=  (/)
).
4 exmid 404 . . . . . . . . . 10  |-  ( ( a  i^i  x )  =  (/)  \/  -.  ( a  i^i  x
)  =  (/) )
5 onfrALTlem1VD 28982 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
65in2an 28685 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( ( a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
7 onfrALTlem2VD 28981 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
87in2an 28685 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( -.  (
a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
9 pm2.61 163 . . . . . . . . . . 11  |-  ( ( ( a  i^i  x
)  =  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  ( ( -.  ( a  i^i  x
)  =  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
109a1i 10 . . . . . . . . . 10  |-  ( ( ( a  i^i  x
)  =  (/)  \/  -.  ( a  i^i  x
)  =  (/) )  -> 
( ( ( a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) )  -> 
( ( -.  (
a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ) )
114, 6, 8, 10e022 28718 . . . . . . . . 9  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  E. y  e.  a  ( a  i^i  y
)  =  (/) ).
1211in2 28682 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( x  e.  a  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
1312gen11 28693 . . . . . . 7  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  A. x
( x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
14 19.23v 1844 . . . . . . . 8  |-  ( A. x ( x  e.  a  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) 
<->  ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) )
1514biimpi 186 . . . . . . 7  |-  ( A. x ( x  e.  a  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) )
1613, 15e1_ 28704 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
17 n0 3477 . . . . . 6  |-  ( a  =/=  (/)  <->  E. x  x  e.  a )
18 imbi1 313 . . . . . . 7  |-  ( ( a  =/=  (/)  <->  E. x  x  e.  a )  ->  ( ( a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) )  <->  ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ) )
1918biimprcd 216 . . . . . 6  |-  ( ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  ( ( a  =/=  (/)  <->  E. x  x  e.  a )  ->  (
a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ) )
2016, 17, 19e10 28772 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
21 pm2.27 35 . . . . 5  |-  ( a  =/=  (/)  ->  ( (
a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
223, 20, 21e11 28765 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
2322in1 28638 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
2423ax-gen 1536 . 2  |-  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) )
25 dfepfr 4394 . . 3  |-  (  _E  Fr  On  <->  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) ) )
2625biimpri 197 . 2  |-  ( A. a ( ( a 
C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  _E  Fr  On )
2724, 26e0_ 28861 1  |-  _E  Fr  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    i^i cin 3164    C_ wss 3165   (/)c0 3468    _E cep 4319    Fr wfr 4365   Oncon0 4408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-vd1 28637  df-vd2 28646  df-vd3 28658
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