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Theorem isf32lem4 8784
Description: Lemma for isfin3-2 8795. Being a chain, difference sets are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a  |-  ( ph  ->  F : om --> ~P G
)
isf32lem.b  |-  ( ph  ->  A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x ) )
isf32lem.c  |-  ( ph  ->  -.  |^| ran  F  e. 
ran  F )
Assertion
Ref Expression
isf32lem4  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( ( ( F `
 A )  \ 
( F `  suc  A ) )  i^i  (
( F `  B
)  \  ( F `  suc  B ) ) )  =  (/) )
Distinct variable groups:    x, B    ph, x    x, A    x, F
Allowed substitution hint:    G( x)

Proof of Theorem isf32lem4
StepHypRef Expression
1 simplrr 769 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  B  e.  om )
2 simplrl 768 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  A  e.  om )
3 simpr 462 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  A  e.  B )
4 simplll 766 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  ph )
5 incom 3661 . . . 4  |-  ( ( ( F `  A
)  \  ( F `  suc  A ) )  i^i  ( ( F `
 B )  \ 
( F `  suc  B ) ) )  =  ( ( ( F `
 B )  \ 
( F `  suc  B ) )  i^i  (
( F `  A
)  \  ( F `  suc  A ) ) )
6 isf32lem.a . . . . 5  |-  ( ph  ->  F : om --> ~P G
)
7 isf32lem.b . . . . 5  |-  ( ph  ->  A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x ) )
8 isf32lem.c . . . . 5  |-  ( ph  ->  -.  |^| ran  F  e. 
ran  F )
96, 7, 8isf32lem3 8783 . . . 4  |-  ( ( ( B  e.  om  /\  A  e.  om )  /\  ( A  e.  B  /\  ph ) )  -> 
( ( ( F `
 B )  \ 
( F `  suc  B ) )  i^i  (
( F `  A
)  \  ( F `  suc  A ) ) )  =  (/) )
105, 9syl5eq 2482 . . 3  |-  ( ( ( B  e.  om  /\  A  e.  om )  /\  ( A  e.  B  /\  ph ) )  -> 
( ( ( F `
 A )  \ 
( F `  suc  A ) )  i^i  (
( F `  B
)  \  ( F `  suc  B ) ) )  =  (/) )
111, 2, 3, 4, 10syl22anc 1265 . 2  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  (
( ( F `  A )  \  ( F `  suc  A ) )  i^i  ( ( F `  B ) 
\  ( F `  suc  B ) ) )  =  (/) )
12 simplrl 768 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  A  e.  om )
13 simplrr 769 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  B  e.  om )
14 simpr 462 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  B  e.  A )
15 simplll 766 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  ph )
166, 7, 8isf32lem3 8783 . . 3  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( ( ( F `
 A )  \ 
( F `  suc  A ) )  i^i  (
( F `  B
)  \  ( F `  suc  B ) ) )  =  (/) )
1712, 13, 14, 15, 16syl22anc 1265 . 2  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  (
( ( F `  A )  \  ( F `  suc  A ) )  i^i  ( ( F `  B ) 
\  ( F `  suc  B ) ) )  =  (/) )
18 simplr 760 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  ->  A  =/=  B )
19 nnord 6714 . . . . . 6  |-  ( A  e.  om  ->  Ord  A )
20 nnord 6714 . . . . . 6  |-  ( B  e.  om  ->  Ord  B )
21 ordtri3 5478 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )
2219, 20, 21syl2an 479 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A
) ) )
2322adantl 467 . . . 4  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A
) ) )
2423necon2abid 2685 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( ( A  e.  B  \/  B  e.  A )  <->  A  =/=  B ) )
2518, 24mpbird 235 . 2  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( A  e.  B  \/  B  e.  A
) )
2611, 17, 25mpjaodan 793 1  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( ( ( F `
 A )  \ 
( F `  suc  A ) )  i^i  (
( F `  B
)  \  ( F `  suc  B ) ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782    \ cdif 3439    i^i cin 3441    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   |^|cint 4258   ran crn 4855   Ord word 5441   suc csuc 5444   -->wf 5597   ` cfv 5601   omcom 6706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fv 5609  df-om 6707
This theorem is referenced by:  isf32lem7  8787
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