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Theorem onfrALTVD 29005
Description: Virtual deduction proof of onfrALT 28637. 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 28637 is onfrALTVD 29005 without virtual deductions and was automatically derived from onfrALTVD 29005.
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 28667 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
2 simpr 449 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  =/=  (/) )
31, 2e1_ 28730 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  a  =/=  (/)
).
4 exmid 406 . . . . . . . . . 10  |-  ( ( a  i^i  x )  =  (/)  \/  -.  ( a  i^i  x
)  =  (/) )
5 onfrALTlem1VD 29004 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
65in2an 28711 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( ( a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
7 onfrALTlem2VD 29003 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
87in2an 28711 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( -.  (
a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
9 pm2.61 166 . . . . . . . . . . 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 11 . . . . . . . . . 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 28744 . . . . . . . . 9  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  E. y  e.  a  ( a  i^i  y
)  =  (/) ).
1211in2 28708 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( x  e.  a  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
1312gen11 28719 . . . . . . 7  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  A. x
( x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
14 19.23v 1915 . . . . . . . 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 188 . . . . . . 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_ 28730 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
17 n0 3639 . . . . . 6  |-  ( a  =/=  (/)  <->  E. x  x  e.  a )
18 imbi1 315 . . . . . . 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 218 . . . . . 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 28797 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
21 pm2.27 38 . . . . 5  |-  ( a  =/=  (/)  ->  ( (
a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
223, 20, 21e11 28791 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
2322in1 28664 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
2423ax-gen 1556 . 2  |-  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) )
25 dfepfr 4569 . . 3  |-  (  _E  Fr  On  <->  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) ) )
2625biimpri 199 . 2  |-  ( A. a ( ( a 
C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  _E  Fr  On )
2724, 26e0_ 28886 1  |-  _E  Fr  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    i^i cin 3321    C_ wss 3322   (/)c0 3630    _E cep 4494    Fr wfr 4540   Oncon0 4583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-vd1 28663  df-vd2 28672  df-vd3 28684
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