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Theorem onfrALTlem2 31606
Description: Lemma for onfrALT 31609. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem2  |-  ( ( 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 onfrALTlem2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( 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
) )
21a1ii 27 . . . . . . . . . . 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  y
) ) ) )
3 inss2 3682 . . . . . . . . . . . 12  |-  ( a  i^i  y )  C_  y
43sseli 3463 . . . . . . . . . . 11  |-  ( z  e.  ( a  i^i  y )  ->  z  e.  y )
52, 4syl8 70 . . . . . . . . . 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.  y ) ) )
6 inss1 3681 . . . . . . . . . . . . 13  |-  ( a  i^i  y )  C_  a
76sseli 3463 . . . . . . . . . . . 12  |-  ( z  e.  ( a  i^i  y )  ->  z  e.  a )
82, 7syl8 70 . . . . . . . . . . 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 ) ) )
9 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  C_  On )
10 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  x  e.  a )
11 ssel 3461 . . . . . . . . . . . . . . 15  |-  ( a 
C_  On  ->  ( x  e.  a  ->  x  e.  On ) )
129, 10, 11syl2im 38 . . . . . . . . . . . . . 14  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  x  e.  On ) )
13 eloni 4840 . . . . . . . . . . . . . 14  |-  ( x  e.  On  ->  Ord  x )
1412, 13syl6 33 . . . . . . . . . . . . 13  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  Ord  x
) )
15 ordtr 4844 . . . . . . . . . . . . 13  |-  ( Ord  x  ->  Tr  x
)
1614, 15syl6 33 . . . . . . . . . . . 12  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  Tr  x
) )
17 simpll 753 . . . . . . . . . . . . . 14  |-  ( ( ( 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
) )
1817a1ii 27 . . . . . . . . . . . . 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 ) )  ->  y  e.  ( a  i^i  x
) ) ) )
19 inss2 3682 . . . . . . . . . . . . . 14  |-  ( a  i^i  x )  C_  x
2019sseli 3463 . . . . . . . . . . . . 13  |-  ( y  e.  ( a  i^i  x )  ->  y  e.  x )
2118, 20syl8 70 . . . . . . . . . . . 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 ) )  ->  y  e.  x ) ) )
22 trel 4503 . . . . . . . . . . . . 13  |-  ( Tr  x  ->  ( (
z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
2322expcomd 438 . . . . . . . . . . . 12  |-  ( Tr  x  ->  ( y  e.  x  ->  ( z  e.  y  ->  z  e.  x ) ) )
2416, 21, 5, 23ee233 31576 . . . . . . . . . . 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.  x ) ) )
25 elin 3650 . . . . . . . . . . . 12  |-  ( z  e.  ( a  i^i  x )  <->  ( z  e.  a  /\  z  e.  x ) )
2625simplbi2 625 . . . . . . . . . . 11  |-  ( z  e.  a  ->  (
z  e.  x  -> 
z  e.  ( a  i^i  x ) ) )
278, 24, 26ee33 31578 . . . . . . . . . 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
) ) ) )
28 elin 3650 . . . . . . . . . . 11  |-  ( z  e.  ( ( a  i^i  x )  i^i  y )  <->  ( z  e.  ( a  i^i  x
)  /\  z  e.  y ) )
2928simplbi2com 627 . . . . . . . . . 10  |-  ( z  e.  y  ->  (
z  e.  ( a  i^i  x )  -> 
z  e.  ( ( a  i^i  x )  i^i  y ) ) )
305, 27, 29ee33 31578 . . . . . . . . 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 )  =  (/) )  /\  z  e.  ( a  i^i  y ) )  ->  z  e.  ( ( a  i^i  x )  i^i  y
) ) ) )
3130exp4a 606 . . . . . . . 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
)  =  (/) )  -> 
( z  e.  ( a  i^i  y )  ->  z  e.  ( ( a  i^i  x
)  i^i  y )
) ) ) )
3231ggen31 31605 . . . . . . 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. z ( z  e.  ( a  i^i  y
)  ->  z  e.  ( ( a  i^i  x )  i^i  y
) ) ) ) )
33 dfss2 3456 . . . . . . . 8  |-  ( ( 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 )
) )
3433biimpri 206 . . . . . . 7  |-  ( 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
) )
3532, 34syl8 70 . . . . . 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
)  =  (/) )  -> 
( a  i^i  y
)  C_  ( (
a  i^i  x )  i^i  y ) ) ) )
36 simpr 461 . . . . . . 7  |-  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
( ( a  i^i  x )  i^i  y
)  =  (/) )
3736a1ii 27 . . . . . 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
)  =  (/) )  -> 
( ( a  i^i  x )  i^i  y
)  =  (/) ) ) )
38 sseq0 3780 . . . . . . 7  |-  ( ( ( a  i^i  y
)  C_  ( (
a  i^i  x )  i^i  y )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) )  ->  (
a  i^i  y )  =  (/) )
3938ex 434 . . . . . 6  |-  ( ( a  i^i  y ) 
C_  ( ( a  i^i  x )  i^i  y )  ->  (
( ( a  i^i  x )  i^i  y
)  =  (/)  ->  (
a  i^i  y )  =  (/) ) )
4035, 37, 39ee33 31578 . . . . 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
)  =  (/) )  -> 
( a  i^i  y
)  =  (/) ) ) )
41 simpl 457 . . . . . . 7  |-  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
y  e.  ( a  i^i  x ) )
4241a1ii 27 . . . . . 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  i^i  x ) ) ) )
43 inss1 3681 . . . . . . 7  |-  ( a  i^i  x )  C_  a
4443sseli 3463 . . . . . 6  |-  ( y  e.  ( a  i^i  x )  ->  y  e.  a )
4542, 44syl8 70 . . . . 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 ) ) )
46 pm3.21 448 . . . . 5  |-  ( ( a  i^i  y )  =  (/)  ->  ( y  e.  a  ->  (
y  e.  a  /\  ( a  i^i  y
)  =  (/) ) ) )
4740, 45, 46ee33 31578 . . . 4  |-  ( ( 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 )  =  (/) ) ) ) )
4847alrimdv 1688 . . 3  |-  ( ( 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 )  =  (/) ) ) ) )
49 onfrALTlem3 31604 . . . 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 )  =  (/) ) )
50 df-rex 2805 . . . 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 )  =  (/) ) )
5149, 50syl6ib 226 . . 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 )  =  (/) ) ) )
52 exim 1624 . . 3  |-  ( 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 )  =  (/) ) ) )
5348, 51, 52syl6c 64 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  E. y
( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ) )
54 df-rex 2805 . 2  |-  ( E. y  e.  a  ( a  i^i  y )  =  (/)  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
5553, 54syl6ibr 227 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 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   E.wrex 2800    i^i cin 3438    C_ wss 3439   (/)c0 3748   Tr wtr 4496   Ord word 4829   Oncon0 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834
This theorem is referenced by:  onfrALT  31609
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