Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onfrALTlem3 Unicode version

Theorem onfrALTlem3 28309
Description: Lemma for onfrALT 28314. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem3  |-  ( ( 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 )  =  (/) ) )
Distinct variable groups:    y, a    x, y

Proof of Theorem onfrALTlem3
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 ssid 3197 . . 3  |-  ( a  i^i  x )  C_  ( a  i^i  x
)
2 simpr 447 . . . . 5  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  -.  ( a  i^i  x
)  =  (/) )
32a1i 10 . . . 4  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  -.  (
a  i^i  x )  =  (/) ) )
4 df-ne 2448 . . . 4  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x )  =  (/) )
53, 4syl6ibr 218 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  ( a  i^i  x )  =/=  (/) ) )
6 pm3.2 434 . . 3  |-  ( ( a  i^i  x ) 
C_  ( a  i^i  x )  ->  (
( a  i^i  x
)  =/=  (/)  ->  (
( a  i^i  x
)  C_  ( a  i^i  x )  /\  (
a  i^i  x )  =/=  (/) ) ) )
71, 5, 6ee02 1367 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  ( ( a  i^i  x ) 
C_  ( a  i^i  x )  /\  (
a  i^i  x )  =/=  (/) ) ) )
8 vex 2791 . . . . 5  |-  x  e. 
_V
98inex2 4156 . . . 4  |-  ( a  i^i  x )  e. 
_V
10 simpl 443 . . . . . . . . . 10  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  C_  On )
11 simpl 443 . . . . . . . . . 10  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  x  e.  a )
12 ssel 3174 . . . . . . . . . 10  |-  ( a 
C_  On  ->  ( x  e.  a  ->  x  e.  On ) )
1310, 11, 12syl2im 34 . . . . . . . . 9  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  x  e.  On ) )
14 eloni 4402 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
1513, 14syl6 29 . . . . . . . 8  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  Ord  x
) )
16 ordwe 4405 . . . . . . . 8  |-  ( Ord  x  ->  _E  We  x )
1715, 16syl6 29 . . . . . . 7  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  _E  We  x ) )
18 inss2 3390 . . . . . . 7  |-  ( a  i^i  x )  C_  x
19 wess 4380 . . . . . . . 8  |-  ( ( a  i^i  x ) 
C_  x  ->  (  _E  We  x  ->  _E  We  ( a  i^i  x
) ) )
2019com12 27 . . . . . . 7  |-  (  _E  We  x  ->  (
( a  i^i  x
)  C_  x  ->  _E  We  ( a  i^i  x ) ) )
2117, 18, 20syl6mpi 58 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  _E  We  ( a  i^i  x
) ) )
22 wefr 4383 . . . . . 6  |-  (  _E  We  ( a  i^i  x )  ->  _E  Fr  ( a  i^i  x
) )
2321, 22syl6 29 . . . . 5  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  _E  Fr  ( a  i^i  x
) ) )
24 dfepfr 4378 . . . . 5  |-  (  _E  Fr  ( a  i^i  x )  <->  A. b
( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) )
2523, 24syl6ib 217 . . . 4  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  A. b
( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) ) )
26 spsbc 3003 . . . 4  |-  ( ( a  i^i  x )  e.  _V  ->  ( A. b ( ( b 
C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) )  ->  [. ( a  i^i  x )  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) ) )
279, 25, 26ee02 1367 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  [. (
a  i^i  x )  /  b ]. (
( b  C_  (
a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b 
( b  i^i  y
)  =  (/) ) ) )
28 onfrALTlem5 28307 . . 3  |-  ( [. ( a  i^i  x
)  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) 
<->  ( ( ( a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  (
a  i^i  x )
( ( a  i^i  x )  i^i  y
)  =  (/) ) )
2927, 28syl6ib 217 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  ( ( ( a  i^i  x
)  C_  ( a  i^i  x )  /\  (
a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x
) ( ( a  i^i  x )  i^i  y )  =  (/) ) ) )
307, 29mpdd 36 1  |-  ( ( 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 )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788   [.wsbc 2991    i^i cin 3151    C_ wss 3152   (/)c0 3455    _E cep 4303    Fr wfr 4349    We wwe 4351   Ord word 4391   Oncon0 4392
This theorem is referenced by:  onfrALTlem2  28311
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
  Copyright terms: Public domain W3C validator