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Theorem isf32lem4 8736
Description: Lemma for isfin3-2 8747. 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 760 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  B  e.  om )
2 simplrl 759 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  A  e.  om )
3 simpr 461 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  A  e.  B )
4 simplll 757 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  ph )
5 incom 3691 . . . 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 8735 . . . 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 2520 . . 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 1229 . 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 759 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  A  e.  om )
13 simplrr 760 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  B  e.  om )
14 simpr 461 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  B  e.  A )
15 simplll 757 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  ph )
166, 7, 8isf32lem3 8735 . . 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 1229 . 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 754 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  ->  A  =/=  B )
19 nnord 6692 . . . . . 6  |-  ( A  e.  om  ->  Ord  A )
20 nnord 6692 . . . . . 6  |-  ( B  e.  om  ->  Ord  B )
21 ordtri3 4914 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )
2219, 20, 21syl2an 477 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A
) ) )
2322adantl 466 . . . 4  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A
) ) )
2423necon2abid 2721 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( ( A  e.  B  \/  B  e.  A )  <->  A  =/=  B ) )
2518, 24mpbird 232 . 2  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( A  e.  B  \/  B  e.  A
) )
2611, 17, 25mpjaodan 784 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 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   |^|cint 4282   Ord word 4877   suc csuc 4880   ran crn 5000   -->wf 5584   ` cfv 5588   omcom 6684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-iota 5551  df-fv 5596  df-om 6685
This theorem is referenced by:  isf32lem7  8739
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