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Theorem isf32lem4 8228
Description: Lemma for isfin3-2 8239. 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 738 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  B  e.  om )
2 simplrl 737 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  A  e.  om )
3 simpr 448 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  A  e.  B )
4 simplll 735 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  ph )
5 incom 3525 . . . 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 8227 . . . 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 2479 . . 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 1185 . 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 737 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  A  e.  om )
13 simplrr 738 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  B  e.  om )
14 simpr 448 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  B  e.  A )
15 simplll 735 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  ph )
166, 7, 8isf32lem3 8227 . . 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 1185 . 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 732 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  ->  A  =/=  B )
19 nnord 4845 . . . . . 6  |-  ( A  e.  om  ->  Ord  A )
20 nnord 4845 . . . . . 6  |-  ( B  e.  om  ->  Ord  B )
21 ordtri3 4609 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )
2219, 20, 21syl2an 464 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A
) ) )
2322adantl 453 . . . 4  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A
) ) )
2423necon2abid 2655 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( ( A  e.  B  \/  B  e.  A )  <->  A  =/=  B ) )
2518, 24mpbird 224 . 2  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( A  e.  B  \/  B  e.  A
) )
2611, 17, 25mpjaodan 762 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   |^|cint 4042   Ord word 4572   suc csuc 4575   omcom 4837   ran crn 4871   -->wf 5442   ` cfv 5446
This theorem is referenced by:  isf32lem7  8231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-iota 5410  df-fv 5454
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