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Theorem onfrALTlem2 28311
Description: Lemma for onfrALT 28314. (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 onfrALTlem3 28309 . . . 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 )  =  (/) ) )
2 df-rex 2549 . . . 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 )  =  (/) ) )
31, 2syl6ib 217 . . 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 )  =  (/) ) ) )
4 simpr 447 . . . . . . . . . . . . . 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
) )
54a1ii 24 . . . . . . . . . . . . 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
) ) ) )
6 inss2 3390 . . . . . . . . . . . . . 14  |-  ( a  i^i  y )  C_  y
76sseli 3176 . . . . . . . . . . . . 13  |-  ( z  e.  ( a  i^i  y )  ->  z  e.  y )
85, 7syl8 65 . . . . . . . . . . . 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 ) ) )
9 inss1 3389 . . . . . . . . . . . . . . 15  |-  ( a  i^i  y )  C_  a
109sseli 3176 . . . . . . . . . . . . . 14  |-  ( z  e.  ( a  i^i  y )  ->  z  e.  a )
115, 10syl8 65 . . . . . . . . . . . . 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 ) ) )
12 simpl 443 . . . . . . . . . . . . . . . . 17  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  C_  On )
13 simpl 443 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  x  e.  a )
14 ssel 3174 . . . . . . . . . . . . . . . . 17  |-  ( a 
C_  On  ->  ( x  e.  a  ->  x  e.  On ) )
1512, 13, 14syl2im 34 . . . . . . . . . . . . . . . 16  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  x  e.  On ) )
16 eloni 4402 . . . . . . . . . . . . . . . 16  |-  ( x  e.  On  ->  Ord  x )
1715, 16syl6 29 . . . . . . . . . . . . . . 15  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  Ord  x
) )
18 ordtr 4406 . . . . . . . . . . . . . . 15  |-  ( Ord  x  ->  Tr  x
)
1917, 18syl6 29 . . . . . . . . . . . . . 14  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  Tr  x
) )
20 simpll 730 . . . . . . . . . . . . . . . 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
) )
2120a1ii 24 . . . . . . . . . . . . . . 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
) ) ) )
22 inss2 3390 . . . . . . . . . . . . . . . 16  |-  ( a  i^i  x )  C_  x
2322sseli 3176 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( a  i^i  x )  ->  y  e.  x )
2421, 23syl8 65 . . . . . . . . . . . . . 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 ) ) )
25 trel 4120 . . . . . . . . . . . . . . 15  |-  ( Tr  x  ->  ( (
z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
2625exp3acom23 1362 . . . . . . . . . . . . . 14  |-  ( Tr  x  ->  ( y  e.  x  ->  ( z  e.  y  ->  z  e.  x ) ) )
2719, 24, 8, 26ee233 28281 . . . . . . . . . . . . 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 ) ) )
28 elin 3358 . . . . . . . . . . . . . 14  |-  ( z  e.  ( a  i^i  x )  <->  ( z  e.  a  /\  z  e.  x ) )
2928simplbi2 608 . . . . . . . . . . . . 13  |-  ( z  e.  a  ->  (
z  e.  x  -> 
z  e.  ( a  i^i  x ) ) )
3011, 27, 29ee33 28284 . . . . . . . . . . . 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
) ) ) )
31 elin 3358 . . . . . . . . . . . . 13  |-  ( z  e.  ( ( a  i^i  x )  i^i  y )  <->  ( z  e.  ( a  i^i  x
)  /\  z  e.  y ) )
3231simplbi2com 1364 . . . . . . . . . . . 12  |-  ( z  e.  y  ->  (
z  e.  ( a  i^i  x )  -> 
z  e.  ( ( a  i^i  x )  i^i  y ) ) )
338, 30, 32ee33 28284 . . . . . . . . . . 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
) ) ) )
3433exp4a 589 . . . . . . . . . 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 )
) ) ) )
3534ggen31 28310 . . . . . . . . 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
) ) ) ) )
36 dfss2 3169 . . . . . . . . . 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 )
) )
3736biimpri 197 . . . . . . . . 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
) )
3835, 37syl8 65 . . . . . . . 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 ) ) ) )
39 simpr 447 . . . . . . . . 9  |-  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
( ( a  i^i  x )  i^i  y
)  =  (/) )
4039a1ii 24 . . . . . . . 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
)  =  (/) ) ) )
41 sseq0 3486 . . . . . . . . 9  |-  ( ( ( a  i^i  y
)  C_  ( (
a  i^i  x )  i^i  y )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) )  ->  (
a  i^i  y )  =  (/) )
4241ex 423 . . . . . . . 8  |-  ( ( a  i^i  y ) 
C_  ( ( a  i^i  x )  i^i  y )  ->  (
( ( a  i^i  x )  i^i  y
)  =  (/)  ->  (
a  i^i  y )  =  (/) ) )
4338, 40, 42ee33 28284 . . . . . . 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
)  =  (/) ) ) )
44 simpl 443 . . . . . . . . 9  |-  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
y  e.  ( a  i^i  x ) )
4544a1ii 24 . . . . . . . 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 ) ) ) )
46 inss1 3389 . . . . . . . . 9  |-  ( a  i^i  x )  C_  a
4746sseli 3176 . . . . . . . 8  |-  ( y  e.  ( a  i^i  x )  ->  y  e.  a )
4845, 47syl8 65 . . . . . . 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 ) ) )
49 pm3.21 435 . . . . . . 7  |-  ( ( a  i^i  y )  =  (/)  ->  ( y  e.  a  ->  (
y  e.  a  /\  ( a  i^i  y
)  =  (/) ) ) )
5043, 48, 49ee33 28284 . . . . . 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 )  =  (/) ) ) ) )
5150idi 2 . . . . 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 )  =  (/) ) ) ) )
5251alrimdv 1619 . . . 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 )  =  (/) ) ) ) )
53 exim 1562 . . . 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 )  =  (/) ) ) )
5452, 53syl6 29 . . 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 )  =  (/) ) ) ) )
553, 54mpdd 36 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  E. y
( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ) )
56 df-rex 2549 . 2  |-  ( E. y  e.  a  ( a  i^i  y )  =  (/)  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
5755, 56syl6ibr 218 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 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   Tr wtr 4113   Ord word 4391   Oncon0 4392
This theorem is referenced by:  onfrALT  28314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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