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Theorem onfrALTlem3VD 28333
Description: Virtual deduction proof of onfrALTlem3 27966. 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 27966 is onfrALTlem3VD 28333 without virtual deductions and was automatically derived from onfrALTlem3VD 28333.
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 2895 . . . . 5  |-  x  e. 
_V
2 inss2 3498 . . . . 5  |-  ( a  i^i  x )  C_  x
31, 2ssexi 4282 . . . 4  |-  ( a  i^i  x )  e. 
_V
4 idn2 28048 . . . . . . . . . . 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 28066 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  x  e.  a ).
7 idn1 27999 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
8 simpl 444 . . . . . . . . . . 11  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  C_  On )
97, 8e1_ 28062 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  a  C_  On ).
10 ssel 3278 . . . . . . . . . . 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 28176 . . . . . . . . 9  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  x  e.  On ).
13 eloni 4525 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
1412, 13e2 28066 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  Ord  x ).
15 ordwe 4528 . . . . . . . 8  |-  ( Ord  x  ->  _E  We  x )
1614, 15e2 28066 . . . . . . 7  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  _E  We  x ).
17 wess 4503 . . . . . . . 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 28173 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  _E  We  ( a  i^i  x
) ).
20 wefr 4506 . . . . . 6  |-  (  _E  We  ( a  i^i  x )  ->  _E  Fr  ( a  i^i  x
) )
2119, 20e2 28066 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  _E  Fr  ( a  i^i  x
) ).
22 dfepfr 4501 . . . . . 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 28066 . . . 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 3109 . . . 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 28132 . . 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 27964 . . 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 28067 . 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 3303 . . 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 28066 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  -.  (
a  i^i  x )  =  (/) ).
32 df-ne 2545 . . . . 5  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x )  =  (/) )
3332biimpri 198 . . . 4  |-  ( -.  ( a  i^i  x
)  =  (/)  ->  (
a  i^i  x )  =/=  (/) )
3431, 33e2 28066 . . 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 28132 . 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 28106 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 1546    = wceq 1649    e. wcel 1717    =/= wne 2543   E.wrex 2643   _Vcvv 2892   [.wsbc 3097    i^i cin 3255    C_ wss 3256   (/)c0 3564    _E cep 4426    Fr wfr 4472    We wwe 4474   Ord word 4514   Oncon0 4515   (.wvd2 28003
This theorem is referenced by:  onfrALTlem2VD  28335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-vd1 27995  df-vd2 28004
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