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Theorem onfrALTlem2VD 37286
Description: Virtual deduction proof of onfrALTlem2 36912. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem2 36912 is onfrALTlem2VD 37286 without virtual deductions and was automatically derived from onfrALTlem2VD 37286.
1::  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) ) ).
2:1:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  ( a  i^i  y ) ).
3:2:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  a ).
4::  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
5::  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ).
6:5:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  x  e.  a ).
7:4:  |-  (. ( a  C_  On  /\  a  =/=  (/) )  ->.  a  C_  On ).
8:6,7:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  x  e.  On ).
9:8:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  Ord  x ).
10:9:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  Tr  x ).
11:1:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  y  e.  ( a  i^i  x ) ).
12:11:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  y  e.  x ).
13:2:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  y ).
14:10,12,13:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  x ).
15:3,14:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  ( a  i^i  x ) ).
16:13,15:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  ( ( a  i^i  x )  i^i  y ) ).
17:16:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y  )  =  (/) )  ->.  ( z  e.  ( a  i^i  y )  ->  z  e.  ( ( a  i^i  x )  i^i  y ) ) ).
18:17:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y  )  =  (/) )  ->.  A. z ( z  e.  ( a  i^i  y )  ->  z  e.  ( ( a  i^i  x )  i^i  y ) ) ).
19:18:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y  )  =  (/) )  ->.  ( a  i^i  y )  C_  ( ( a  i^i  x )  i^i  y ) ).
20::  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y  )  =  (/) )  ->.  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) ).
21:20:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y  )  =  (/) )  ->.  ( ( a  i^i  x )  i^i  y )  =  (/) ).
22:19,21:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y  )  =  (/) )  ->.  ( a  i^i  y )  =  (/) ).
23:20:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y  )  =  (/) )  ->.  y  e.  ( a  i^i  x ) ).
24:23:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y  )  =  (/) )  ->.  y  e.  a ).
25:22,24:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) ) ,  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y  )  =  (/) )  ->.  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ).
26:25:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  ->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ) ).
27:26:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  A. y ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x  )  i^i  y )  =  (/) )  ->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ) ).
28:27:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  ( E. y ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x  )  i^i  y )  =  (/) )  ->  E. y ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ) ).
29::  |-  (. ( 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  )  =  (/) ).
30:29:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  E. y ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) ).
31:28,30:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  E. y ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ).
qed:31:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->.  E. y  e.  a ( a  i^i  y )  =  (/) ).
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem2VD  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
Distinct variable groups:    y, a    x, y

Proof of Theorem onfrALTlem2VD
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 idn3 36994 . . . . . . . . . . . . . 14  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  ( ( y  e.  ( a  i^i  x )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y
) ) ).
2 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->  z  e.  ( a  i^i  y
) )
31, 2e3 37124 . . . . . . . . . . . . 13  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  ( a  i^i  y ) ).
4 inss2 3653 . . . . . . . . . . . . . 14  |-  ( a  i^i  y )  C_  y
54sseli 3428 . . . . . . . . . . . . 13  |-  ( z  e.  ( a  i^i  y )  ->  z  e.  y )
63, 5e3 37124 . . . . . . . . . . . 12  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  y ).
7 inss1 3652 . . . . . . . . . . . . . . 15  |-  ( a  i^i  y )  C_  a
87sseli 3428 . . . . . . . . . . . . . 14  |-  ( z  e.  ( a  i^i  y )  ->  z  e.  a )
93, 8e3 37124 . . . . . . . . . . . . 13  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  a ).
10 idn2 36992 . . . . . . . . . . . . . . . . . 18  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  ( x  e.  a  /\  -.  (
a  i^i  x )  =  (/) ) ).
11 simpl 459 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  x  e.  a )
1210, 11e2 37010 . . . . . . . . . . . . . . . . 17  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  x  e.  a ).
13 idn1 36944 . . . . . . . . . . . . . . . . . 18  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
14 simpl 459 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  C_  On )
1513, 14e1a 37006 . . . . . . . . . . . . . . . . 17  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  a  C_  On ).
16 ssel 3426 . . . . . . . . . . . . . . . . . 18  |-  ( a 
C_  On  ->  ( x  e.  a  ->  x  e.  On ) )
1716com12 32 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  a  ->  (
a  C_  On  ->  x  e.  On ) )
1812, 15, 17e21 37117 . . . . . . . . . . . . . . . 16  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  x  e.  On ).
19 eloni 5433 . . . . . . . . . . . . . . . 16  |-  ( x  e.  On  ->  Ord  x )
2018, 19e2 37010 . . . . . . . . . . . . . . 15  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  Ord  x ).
21 ordtr 5437 . . . . . . . . . . . . . . 15  |-  ( Ord  x  ->  Tr  x
)
2220, 21e2 37010 . . . . . . . . . . . . . 14  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  Tr  x ).
23 simpll 760 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->  y  e.  ( a  i^i  x
) )
241, 23e3 37124 . . . . . . . . . . . . . . 15  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  y  e.  ( a  i^i  x ) ).
25 inss2 3653 . . . . . . . . . . . . . . . 16  |-  ( a  i^i  x )  C_  x
2625sseli 3428 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( a  i^i  x )  ->  y  e.  x )
2724, 26e3 37124 . . . . . . . . . . . . . 14  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  y  e.  x ).
28 trel 4504 . . . . . . . . . . . . . . 15  |-  ( Tr  x  ->  ( (
z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
2928expcomd 440 . . . . . . . . . . . . . 14  |-  ( Tr  x  ->  ( y  e.  x  ->  ( z  e.  y  ->  z  e.  x ) ) )
3022, 27, 6, 29e233 37152 . . . . . . . . . . . . 13  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  x ).
31 elin 3617 . . . . . . . . . . . . . 14  |-  ( z  e.  ( a  i^i  x )  <->  ( z  e.  a  /\  z  e.  x ) )
3231simplbi2 631 . . . . . . . . . . . . 13  |-  ( z  e.  a  ->  (
z  e.  x  -> 
z  e.  ( a  i^i  x ) ) )
339, 30, 32e33 37121 . . . . . . . . . . . 12  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  ( a  i^i  x ) ).
34 elin 3617 . . . . . . . . . . . . 13  |-  ( z  e.  ( ( a  i^i  x )  i^i  y )  <->  ( z  e.  ( a  i^i  x
)  /\  z  e.  y ) )
3534simplbi2com 633 . . . . . . . . . . . 12  |-  ( z  e.  y  ->  (
z  e.  ( a  i^i  x )  -> 
z  e.  ( ( a  i^i  x )  i^i  y ) ) )
366, 33, 35e33 37121 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->.  z  e.  ( ( a  i^i  x
)  i^i  y ) ).
3736in3an 36990 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) 
->.  ( z  e.  ( a  i^i  y )  ->  z  e.  ( ( a  i^i  x
)  i^i  y )
) ).
3837gen31 37000 . . . . . . . . 9  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) 
->.  A. z ( z  e.  ( a  i^i  y )  ->  z  e.  ( ( a  i^i  x )  i^i  y
) ) ).
39 dfss2 3421 . . . . . . . . . 10  |-  ( ( a  i^i  y ) 
C_  ( ( a  i^i  x )  i^i  y )  <->  A. z
( z  e.  ( a  i^i  y )  ->  z  e.  ( ( a  i^i  x
)  i^i  y )
) )
4039biimpri 210 . . . . . . . . 9  |-  ( A. z ( z  e.  ( a  i^i  y
)  ->  z  e.  ( ( a  i^i  x )  i^i  y
) )  ->  (
a  i^i  y )  C_  ( ( a  i^i  x )  i^i  y
) )
4138, 40e3 37124 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) 
->.  ( a  i^i  y
)  C_  ( (
a  i^i  x )  i^i  y ) ).
42 idn3 36994 . . . . . . . . 9  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) 
->.  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) ).
43 simpr 463 . . . . . . . . 9  |-  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
( ( a  i^i  x )  i^i  y
)  =  (/) )
4442, 43e3 37124 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) 
->.  ( ( a  i^i  x )  i^i  y
)  =  (/) ).
45 sseq0 3766 . . . . . . . . 9  |-  ( ( ( a  i^i  y
)  C_  ( (
a  i^i  x )  i^i  y )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) )  ->  (
a  i^i  y )  =  (/) )
4645ex 436 . . . . . . . 8  |-  ( ( a  i^i  y ) 
C_  ( ( a  i^i  x )  i^i  y )  ->  (
( ( a  i^i  x )  i^i  y
)  =  (/)  ->  (
a  i^i  y )  =  (/) ) )
4741, 44, 46e33 37121 . . . . . . 7  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) 
->.  ( a  i^i  y
)  =  (/) ).
48 simpl 459 . . . . . . . . 9  |-  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
y  e.  ( a  i^i  x ) )
4942, 48e3 37124 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) 
->.  y  e.  (
a  i^i  x ) ).
50 inss1 3652 . . . . . . . . 9  |-  ( a  i^i  x )  C_  a
5150sseli 3428 . . . . . . . 8  |-  ( y  e.  ( a  i^i  x )  ->  y  e.  a )
5249, 51e3 37124 . . . . . . 7  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) 
->.  y  e.  a ).
53 pm3.21 450 . . . . . . 7  |-  ( ( a  i^i  y )  =  (/)  ->  ( y  e.  a  ->  (
y  e.  a  /\  ( a  i^i  y
)  =  (/) ) ) )
5447, 52, 53e33 37121 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) ) ,. ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) 
->.  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ).
5554in3 36988 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  ( (
y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ) ).
5655gen21 36998 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  A. y
( ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  ->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ) ).
57 exim 1706 . . . 4  |-  ( A. y ( ( y  e.  ( a  i^i  x )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) )  ->  (
y  e.  a  /\  ( a  i^i  y
)  =  (/) ) )  ->  ( E. y
( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) )  ->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) ) )
5856, 57e2 37010 . . 3  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  ( E. y ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  ->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) ) ).
59 onfrALTlem3VD 37284 . . . 4  |-  (. (
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 )  =  (/) ).
60 df-rex 2743 . . . . 5  |-  ( E. y  e.  ( a  i^i  x ) ( ( a  i^i  x
)  i^i  y )  =  (/)  <->  E. y ( y  e.  ( a  i^i  x )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) ) )
6160biimpi 198 . . . 4  |-  ( E. y  e.  ( a  i^i  x ) ( ( a  i^i  x
)  i^i  y )  =  (/)  ->  E. y
( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
6259, 61e2 37010 . . 3  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y
( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) ).
63 id 22 . . 3  |-  ( ( E. y ( y  e.  ( a  i^i  x )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) )  ->  E. y
( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )  ->  ( E. y ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) )  ->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) ) )
6458, 62, 63e22 37050 . 2  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y
( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ).
65 df-rex 2743 . . 3  |-  ( E. y  e.  a  ( a  i^i  y )  =  (/)  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
6665biimpri 210 . 2  |-  ( E. y ( y  e.  a  /\  ( a  i^i  y )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
6764, 66e2 37010 1  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   E.wrex 2738    i^i cin 3403    C_ wss 3404   (/)c0 3731   Tr wtr 4497   Ord word 5422   Oncon0 5423   (.wvd2 36947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426  df-on 5427  df-vd1 36940  df-vd2 36948  df-vd3 36960
This theorem is referenced by:  onfrALTVD  37288
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