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Theorem onfrALTlem2VD 29075
Description: Virtual deduction proof of onfrALTlem2 28706. 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. onfrALTlem2 28706 is onfrALTlem2VD 29075 without virtual deductions and was automatically derived from onfrALTlem2VD 29075.
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 28790 . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . 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 28923 . . . . . . . . . . . . 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 3564 . . . . . . . . . . . . . 14  |-  ( a  i^i  y )  C_  y
54sseli 3346 . . . . . . . . . . . . 13  |-  ( z  e.  ( a  i^i  y )  ->  z  e.  y )
63, 5e3 28923 . . . . . . . . . . . 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 3563 . . . . . . . . . . . . . . 15  |-  ( a  i^i  y )  C_  a
87sseli 3346 . . . . . . . . . . . . . 14  |-  ( z  e.  ( a  i^i  y )  ->  z  e.  a )
93, 8e3 28923 . . . . . . . . . . . . 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 28788 . . . . . . . . . . . . . . . . . 18  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  ( x  e.  a  /\  -.  (
a  i^i  x )  =  (/) ) ).
11 simpl 445 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  x  e.  a )
1210, 11e2 28806 . . . . . . . . . . . . . . . . 17  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  x  e.  a ).
13 idn1 28739 . . . . . . . . . . . . . . . . . 18  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
14 simpl 445 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  C_  On )
1513, 14e1_ 28802 . . . . . . . . . . . . . . . . 17  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  a  C_  On ).
16 ssel 3344 . . . . . . . . . . . . . . . . . 18  |-  ( a 
C_  On  ->  ( x  e.  a  ->  x  e.  On ) )
1716com12 30 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  a  ->  (
a  C_  On  ->  x  e.  On ) )
1812, 15, 17e21 28916 . . . . . . . . . . . . . . . 16  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  x  e.  On ).
19 eloni 4594 . . . . . . . . . . . . . . . 16  |-  ( x  e.  On  ->  Ord  x )
2018, 19e2 28806 . . . . . . . . . . . . . . 15  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  Ord  x ).
21 ordtr 4598 . . . . . . . . . . . . . . 15  |-  ( Ord  x  ->  Tr  x
)
2220, 21e2 28806 . . . . . . . . . . . . . 14  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  Tr  x ).
23 simpll 732 . . . . . . . . . . . . . . . 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 28923 . . . . . . . . . . . . . . 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 3564 . . . . . . . . . . . . . . . 16  |-  ( a  i^i  x )  C_  x
2625sseli 3346 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( a  i^i  x )  ->  y  e.  x )
2724, 26e3 28923 . . . . . . . . . . . . . 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 4312 . . . . . . . . . . . . . . 15  |-  ( Tr  x  ->  ( (
z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
2928exp3acom23 1382 . . . . . . . . . . . . . 14  |-  ( Tr  x  ->  ( y  e.  x  ->  ( z  e.  y  ->  z  e.  x ) ) )
3022, 27, 6, 29e233 28951 . . . . . . . . . . . . 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 3532 . . . . . . . . . . . . . 14  |-  ( z  e.  ( a  i^i  x )  <->  ( z  e.  a  /\  z  e.  x ) )
3231simplbi2 610 . . . . . . . . . . . . 13  |-  ( z  e.  a  ->  (
z  e.  x  -> 
z  e.  ( a  i^i  x ) ) )
339, 30, 32e33 28920 . . . . . . . . . . . 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 3532 . . . . . . . . . . . . 13  |-  ( z  e.  ( ( a  i^i  x )  i^i  y )  <->  ( z  e.  ( a  i^i  x
)  /\  z  e.  y ) )
3534simplbi2com 1384 . . . . . . . . . . . 12  |-  ( z  e.  y  ->  (
z  e.  ( a  i^i  x )  -> 
z  e.  ( ( a  i^i  x )  i^i  y ) ) )
366, 33, 35e33 28920 . . . . . . . . . . 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 28786 . . . . . . . . . 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 28796 . . . . . . . . 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 3339 . . . . . . . . . 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 199 . . . . . . . . 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 28923 . . . . . . . 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 28790 . . . . . . . . 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 449 . . . . . . . . 9  |-  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
( ( a  i^i  x )  i^i  y
)  =  (/) )
4442, 43e3 28923 . . . . . . . 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 3661 . . . . . . . . 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 425 . . . . . . . 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 28920 . . . . . . 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 445 . . . . . . . . 9  |-  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
y  e.  ( a  i^i  x ) )
4942, 48e3 28923 . . . . . . . 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 3563 . . . . . . . . 9  |-  ( a  i^i  x )  C_  a
5150sseli 3346 . . . . . . . 8  |-  ( y  e.  ( a  i^i  x )  ->  y  e.  a )
5249, 51e3 28923 . . . . . . 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 pm3.21 437 . . . . . . 7  |-  ( ( a  i^i  y )  =  (/)  ->  ( y  e.  a  ->  (
y  e.  a  /\  ( a  i^i  y
)  =  (/) ) ) )
5447, 52, 53e33 28920 . . . . . 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 28784 . . . . 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 28794 . . . 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 1585 . . . 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 28806 . . 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 29073 . . . 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 2713 . . . . 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 188 . . . 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 28806 . . 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 21 . . 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 28846 . 2  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y
( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ).
65 df-rex 2713 . . 3  |-  ( E. y  e.  a  ( a  i^i  y )  =  (/)  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
6665biimpri 199 . 2  |-  ( E. y ( y  e.  a  /\  ( a  i^i  y )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
6764, 66e2 28806 1  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ 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   Tr wtr 4305   Ord word 4583   Oncon0 4584   (.wvd2 28743
This theorem is referenced by:  onfrALTVD  29077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pr 4406
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 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-vd1 28735  df-vd2 28744  df-vd3 28756
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