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Theorem onfrALTlem3VD 28979
Description: Virtual deduction proof of onfrALTlem3 28608. 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 28608 is onfrALTlem3VD 28979 without virtual deductions and was automatically derived from onfrALTlem3VD 28979.
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 2804 . . . . 5  |-  x  e. 
_V
2 inss2 3403 . . . . 5  |-  ( a  i^i  x )  C_  x
31, 2ssexi 4175 . . . 4  |-  ( a  i^i  x )  e. 
_V
4 idn2 28690 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  ( x  e.  a  /\  -.  (
a  i^i  x )  =  (/) ) ).
5 simpl 443 . . . . . . . . . . 11  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  x  e.  a )
64, 5e2 28708 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  x  e.  a ).
7 idn1 28641 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
8 simpl 443 . . . . . . . . . . 11  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  C_  On )
97, 8e1_ 28704 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  a  C_  On ).
10 ssel 3187 . . . . . . . . . . 11  |-  ( a 
C_  On  ->  ( x  e.  a  ->  x  e.  On ) )
1110com12 27 . . . . . . . . . 10  |-  ( x  e.  a  ->  (
a  C_  On  ->  x  e.  On ) )
126, 9, 11e21 28819 . . . . . . . . 9  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  x  e.  On ).
13 eloni 4418 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
1412, 13e2 28708 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  Ord  x ).
15 ordwe 4421 . . . . . . . 8  |-  ( Ord  x  ->  _E  We  x )
1614, 15e2 28708 . . . . . . 7  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  _E  We  x ).
17 wess 4396 . . . . . . . 8  |-  ( ( a  i^i  x ) 
C_  x  ->  (  _E  We  x  ->  _E  We  ( a  i^i  x
) ) )
1817com12 27 . . . . . . 7  |-  (  _E  We  x  ->  (
( a  i^i  x
)  C_  x  ->  _E  We  ( a  i^i  x ) ) )
1916, 2, 18e20 28816 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  _E  We  ( a  i^i  x
) ).
20 wefr 4399 . . . . . 6  |-  (  _E  We  ( a  i^i  x )  ->  _E  Fr  ( a  i^i  x
) )
2119, 20e2 28708 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  _E  Fr  ( a  i^i  x
) ).
22 dfepfr 4394 . . . . . 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 186 . . . . 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 28708 . . . 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 3016 . . . 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 28775 . . 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 28606 . . 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 28709 . 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 3210 . . 3  |-  ( a  i^i  x )  C_  ( a  i^i  x
)
30 simpr 447 . . . . 5  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  -.  ( a  i^i  x
)  =  (/) )
314, 30e2 28708 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  -.  (
a  i^i  x )  =  (/) ).
32 df-ne 2461 . . . . 5  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x )  =  (/) )
3332biimpri 197 . . . 4  |-  ( -.  ( a  i^i  x
)  =  (/)  ->  (
a  i^i  x )  =/=  (/) )
3431, 33e2 28708 . . 3  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  ( a  i^i  x )  =/=  (/) ).
35 pm3.2 434 . . 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 28775 . 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 19 . 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 28748 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 358   A.wal 1530    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   _Vcvv 2801   [.wsbc 3004    i^i cin 3164    C_ wss 3165   (/)c0 3468    _E cep 4319    Fr wfr 4365    We wwe 4367   Ord word 4407   Oncon0 4408   (.wvd2 28645
This theorem is referenced by:  onfrALTlem2VD  28981
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
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