Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onfrALTlem2 Structured version   Unicode version

Theorem onfrALTlem2 28694
Description: Lemma for onfrALT 28697. (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 28692 . . . 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 2713 . . . 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 219 . . 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 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
) )
54a1ii 26 . . . . . . . . . . . . 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 3564 . . . . . . . . . . . . . 14  |-  ( a  i^i  y )  C_  y
76sseli 3346 . . . . . . . . . . . . 13  |-  ( z  e.  ( a  i^i  y )  ->  z  e.  y )
85, 7syl8 68 . . . . . . . . . . . 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 3563 . . . . . . . . . . . . . . 15  |-  ( a  i^i  y )  C_  a
109sseli 3346 . . . . . . . . . . . . . 14  |-  ( z  e.  ( a  i^i  y )  ->  z  e.  a )
115, 10syl8 68 . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . 17  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  C_  On )
13 simpl 445 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  x  e.  a )
14 ssel 3344 . . . . . . . . . . . . . . . . 17  |-  ( a 
C_  On  ->  ( x  e.  a  ->  x  e.  On ) )
1512, 13, 14syl2im 37 . . . . . . . . . . . . . . . 16  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  x  e.  On ) )
16 eloni 4593 . . . . . . . . . . . . . . . 16  |-  ( x  e.  On  ->  Ord  x )
1715, 16syl6 32 . . . . . . . . . . . . . . 15  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  Ord  x
) )
18 ordtr 4597 . . . . . . . . . . . . . . 15  |-  ( Ord  x  ->  Tr  x
)
1917, 18syl6 32 . . . . . . . . . . . . . 14  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  Tr  x
) )
20 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
) )
2120a1ii 26 . . . . . . . . . . . . . . 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 3564 . . . . . . . . . . . . . . . 16  |-  ( a  i^i  x )  C_  x
2322sseli 3346 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( a  i^i  x )  ->  y  e.  x )
2421, 23syl8 68 . . . . . . . . . . . . . 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 4311 . . . . . . . . . . . . . . 15  |-  ( Tr  x  ->  ( (
z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
2625exp3acom23 1382 . . . . . . . . . . . . . 14  |-  ( Tr  x  ->  ( y  e.  x  ->  ( z  e.  y  ->  z  e.  x ) ) )
2719, 24, 8, 26ee233 28664 . . . . . . . . . . . . 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 3532 . . . . . . . . . . . . . 14  |-  ( z  e.  ( a  i^i  x )  <->  ( z  e.  a  /\  z  e.  x ) )
2928simplbi2 610 . . . . . . . . . . . . 13  |-  ( z  e.  a  ->  (
z  e.  x  -> 
z  e.  ( a  i^i  x ) ) )
3011, 27, 29ee33 28667 . . . . . . . . . . . 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 3532 . . . . . . . . . . . . 13  |-  ( z  e.  ( ( a  i^i  x )  i^i  y )  <->  ( z  e.  ( a  i^i  x
)  /\  z  e.  y ) )
3231simplbi2com 1384 . . . . . . . . . . . 12  |-  ( z  e.  y  ->  (
z  e.  ( a  i^i  x )  -> 
z  e.  ( ( a  i^i  x )  i^i  y ) ) )
338, 30, 32ee33 28667 . . . . . . . . . . 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 591 . . . . . . . . . 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 28693 . . . . . . . . 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 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 )
) )
3736biimpri 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
) )
3835, 37syl8 68 . . . . . . . 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 449 . . . . . . . . 9  |-  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
( ( a  i^i  x )  i^i  y
)  =  (/) )
4039a1ii 26 . . . . . . . 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 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 )  =  (/) )
4241ex 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 )  =  (/) ) )
4338, 40, 42ee33 28667 . . . . . . 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 445 . . . . . . . . 9  |-  ( ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y
)  =  (/) )  -> 
y  e.  ( a  i^i  x ) )
4544a1ii 26 . . . . . . . 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 3563 . . . . . . . . 9  |-  ( a  i^i  x )  C_  a
4746sseli 3346 . . . . . . . 8  |-  ( y  e.  ( a  i^i  x )  ->  y  e.  a )
4845, 47syl8 68 . . . . . . 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 pm3.21 437 . . . . . . 7  |-  ( ( a  i^i  y )  =  (/)  ->  ( y  e.  a  ->  (
y  e.  a  /\  ( a  i^i  y
)  =  (/) ) ) )
5043, 48, 49ee33 28667 . . . . . 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 1644 . . . 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 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 )  =  (/) ) ) )
5452, 53syl6 32 . . 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 39 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  E. y
( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ) )
56 df-rex 2713 . 2  |-  ( E. y  e.  a  ( a  i^i  y )  =  (/)  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
5755, 56syl6ibr 220 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 4304   Ord word 4582   Oncon0 4583
This theorem is referenced by:  onfrALT  28697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pr 4405
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 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587
  Copyright terms: Public domain W3C validator