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Theorem onfrALTlem3 36980
Description: Lemma for onfrALT 36985. (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 3437 . . 3  |-  ( a  i^i  x )  C_  ( a  i^i  x
)
2 simpr 468 . . . . 5  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  -.  ( a  i^i  x
)  =  (/) )
32a1i 11 . . . 4  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  -.  (
a  i^i  x )  =  (/) ) )
4 df-ne 2643 . . . 4  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x )  =  (/) )
53, 4syl6ibr 235 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  ( a  i^i  x )  =/=  (/) ) )
6 pm3.2 454 . . 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 36906 . 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 3034 . . . . 5  |-  x  e. 
_V
98inex2 4538 . . . 4  |-  ( a  i^i  x )  e. 
_V
10 inss2 3644 . . . . . . 7  |-  ( a  i^i  x )  C_  x
11 simpl 464 . . . . . . . . . 10  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  C_  On )
12 simpl 464 . . . . . . . . . 10  |-  ( ( x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->  x  e.  a )
13 ssel 3412 . . . . . . . . . 10  |-  ( a 
C_  On  ->  ( x  e.  a  ->  x  e.  On ) )
1411, 12, 13syl2im 38 . . . . . . . . 9  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  x  e.  On ) )
15 eloni 5440 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
1614, 15syl6 33 . . . . . . . 8  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  Ord  x
) )
17 ordwe 5443 . . . . . . . 8  |-  ( Ord  x  ->  _E  We  x )
1816, 17syl6 33 . . . . . . 7  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  _E  We  x ) )
19 wess 4826 . . . . . . 7  |-  ( ( a  i^i  x ) 
C_  x  ->  (  _E  We  x  ->  _E  We  ( a  i^i  x
) ) )
2010, 18, 19ee02 36906 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  _E  We  ( a  i^i  x
) ) )
21 wefr 4829 . . . . . 6  |-  (  _E  We  ( a  i^i  x )  ->  _E  Fr  ( a  i^i  x
) )
2220, 21syl6 33 . . . . 5  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  _E  Fr  ( a  i^i  x
) ) )
23 dfepfr 4824 . . . . 5  |-  (  _E  Fr  ( a  i^i  x )  <->  A. b
( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) )
2422, 23syl6ib 234 . . . 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 )  =  (/) ) ) )
25 spsbc 3268 . . . 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 )  =  (/) ) ) )
269, 24, 25ee02 36906 . . 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
)  =  (/) ) ) )
27 onfrALTlem5 36978 . . 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
)  =  (/) ) )
2826, 27syl6ib 234 . 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 )  =  (/) ) ) )
297, 28mpdd 40 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376   A.wal 1450    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031   [.wsbc 3255    i^i cin 3389    C_ wss 3390   (/)c0 3722    _E cep 4748    Fr wfr 4795    We wwe 4797   Ord word 5429   Oncon0 5430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-ord 5433  df-on 5434
This theorem is referenced by:  onfrALTlem2  36982
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