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Theorem onfrALTlem3VD 28936
Description: Virtual deduction proof of onfrALTlem3 28567. 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. onfrALTlem3 28567 is onfrALTlem3VD 28936 without virtual deductions and was automatically derived from onfrALTlem3VD 28936.
1::  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
2::  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ).
3:2:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  x  e.  a ).
4:1:  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  a  C_  On ).
5:3,4:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  x  e.  On ).
6:5:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  Ord  x ).
7:6:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  _E  We  x ).
8::  |-  ( a  i^i  x )  C_  x
9:7,8:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  _E  We  ( a  i^i  x ) ).
10:9:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  _E  Fr  ( a  i^i  x ) ).
11:10:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  A. b ( ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b ( b  i^i  y )  =  (/) ) ).
12::  |-  x  e.  _V
13:12,8:  |-  ( a  i^i  x )  e.  _V
14:13,11:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  [. ( a  i^i  x )  /  b ]. ( ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b ( b  i^i  y )  =  (/) ) ).
15::  |-  ( [. ( a  i^i  x )  /  b ]. ( ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b ( b  i^i  y )  =  (/) )  <->  ( ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x ) (  ( a  i^i  x )  i^i  y )  =  (/) ) )
16:14,15:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  ( ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  (  a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) ) ).
17::  |-  ( a  i^i  x )  C_  ( a  i^i  x )
18:2:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  -.  ( a  i^i  x )  =  (/) ).
19:18:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  ( a  i^i  x )  =/=  (/) ).
20:17,19:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) ) ).
qed:16,20:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y  )  =  (/) ).
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem3VD  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  ( a  i^i  x
) ( ( a  i^i  x )  i^i  y )  =  (/) ).
Distinct variable groups:    y, a    x, y

Proof of Theorem onfrALTlem3VD
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 vex 2951 . . . . 5  |-  x  e. 
_V
2 inss2 3554 . . . . 5  |-  ( a  i^i  x )  C_  x
31, 2ssexi 4340 . . . 4  |-  ( a  i^i  x )  e. 
_V
4 idn2 28651 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  ( x  e.  a  /\  -.  (
a  i^i  x )  =  (/) ) ).
5 simpl 444 . . . . . . . . . . 11  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  x  e.  a )
64, 5e2 28669 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  x  e.  a ).
7 idn1 28602 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
8 simpl 444 . . . . . . . . . . 11  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  C_  On )
97, 8e1_ 28665 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  a  C_  On ).
10 ssel 3334 . . . . . . . . . . 11  |-  ( a 
C_  On  ->  ( x  e.  a  ->  x  e.  On ) )
1110com12 29 . . . . . . . . . 10  |-  ( x  e.  a  ->  (
a  C_  On  ->  x  e.  On ) )
126, 9, 11e21 28779 . . . . . . . . 9  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  x  e.  On ).
13 eloni 4583 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
1412, 13e2 28669 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  Ord  x ).
15 ordwe 4586 . . . . . . . 8  |-  ( Ord  x  ->  _E  We  x )
1614, 15e2 28669 . . . . . . 7  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  _E  We  x ).
17 wess 4561 . . . . . . . 8  |-  ( ( a  i^i  x ) 
C_  x  ->  (  _E  We  x  ->  _E  We  ( a  i^i  x
) ) )
1817com12 29 . . . . . . 7  |-  (  _E  We  x  ->  (
( a  i^i  x
)  C_  x  ->  _E  We  ( a  i^i  x ) ) )
1916, 2, 18e20 28776 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  _E  We  ( a  i^i  x
) ).
20 wefr 4564 . . . . . 6  |-  (  _E  We  ( a  i^i  x )  ->  _E  Fr  ( a  i^i  x
) )
2119, 20e2 28669 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  _E  Fr  ( a  i^i  x
) ).
22 dfepfr 4559 . . . . . 6  |-  (  _E  Fr  ( a  i^i  x )  <->  A. b
( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) )
2322biimpi 187 . . . . 5  |-  (  _E  Fr  ( a  i^i  x )  ->  A. b
( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) )
2421, 23e2 28669 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  A. b
( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) ).
25 spsbc 3165 . . . 4  |-  ( ( a  i^i  x )  e.  _V  ->  ( A. b ( ( b 
C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) )  ->  [. ( a  i^i  x )  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) ) )
263, 24, 25e02 28735 . . 3  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  [. ( a  i^i  x )  / 
b ]. ( ( b 
C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) ).
27 onfrALTlem5 28565 . . 3  |-  ( [. ( a  i^i  x
)  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) 
<->  ( ( ( a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  (
a  i^i  x )
( ( a  i^i  x )  i^i  y
)  =  (/) ) )
2826, 27e2bi 28670 . 2  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  ( (
( a  i^i  x
)  C_  ( a  i^i  x )  /\  (
a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x
) ( ( a  i^i  x )  i^i  y )  =  (/) ) ).
29 ssid 3359 . . 3  |-  ( a  i^i  x )  C_  ( a  i^i  x
)
30 simpr 448 . . . . 5  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  -.  ( a  i^i  x
)  =  (/) )
314, 30e2 28669 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  -.  (
a  i^i  x )  =  (/) ).
32 df-ne 2600 . . . . 5  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x )  =  (/) )
3332biimpri 198 . . . 4  |-  ( -.  ( a  i^i  x
)  =  (/)  ->  (
a  i^i  x )  =/=  (/) )
3431, 33e2 28669 . . 3  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  ( a  i^i  x )  =/=  (/) ).
35 pm3.2 435 . . 3  |-  ( ( a  i^i  x ) 
C_  ( a  i^i  x )  ->  (
( a  i^i  x
)  =/=  (/)  ->  (
( a  i^i  x
)  C_  ( a  i^i  x )  /\  (
a  i^i  x )  =/=  (/) ) ) )
3629, 34, 35e02 28735 . 2  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  ( (
a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) ) ).
37 id 20 . 2  |-  ( ( ( ( a  i^i  x )  C_  (
a  i^i  x )  /\  ( a  i^i  x
)  =/=  (/) )  ->  E. y  e.  (
a  i^i  x )
( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
( ( ( a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  (
a  i^i  x )
( ( a  i^i  x )  i^i  y
)  =  (/) ) )
3828, 36, 37e22 28709 1  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  ( a  i^i  x
) ( ( a  i^i  x )  i^i  y )  =  (/) ).
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   _Vcvv 2948   [.wsbc 3153    i^i cin 3311    C_ wss 3312   (/)c0 3620    _E cep 4484    Fr wfr 4530    We wwe 4532   Ord word 4572   Oncon0 4573   (.wvd2 28606
This theorem is referenced by:  onfrALTlem2VD  28938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-vd1 28598  df-vd2 28607
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