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Theorem isf32lem4 9178
Description: Lemma for isfin3-2 9189. 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 801 . . 3 |- ( ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) /\ A e. B ) -> B e. om )
2 simplrl 800 . . 3 |- ( ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) /\ A e. B ) -> A e. om )
3 simpr 477 . . 3 |- ( ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) /\ A e. B ) -> A e. B )
4 simplll 798 . . 3 |- ( ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) /\ A e. B ) -> ph )
5 incom 3805 . . . 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 9177 . . . 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 2668 . . 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 1327 . 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 800 . . 3 |- ( ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) /\ B e. A ) -> A e. om )
13 simplrr 801 . . 3 |- ( ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) /\ B e. A ) -> B e. om )
14 simpr 477 . . 3 |- ( ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) /\ B e. A ) -> B e. A )
15 simplll 798 . . 3 |- ( ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) /\ B e. A ) -> ph )
166, 7, 8isf32lem3 9177 . . 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 1327 . 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 792 . . 3 |- ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) -> A =/= B )
19 nnord 7073 . . . . . 6 |- ( A e. om -> Ord A )
20 nnord 7073 . . . . . 6 |- ( B e. om -> Ord B )
21 ordtri3 5759 . . . . . 6 |- ( ( Ord A /\ Ord B ) -> ( A = B <-> -. ( A e. B \/ B e. A ) ) )
2219, 20, 21syl2an 494 . . . . 5 |- ( ( A e. om /\ B e. om ) -> ( A = B <-> -. ( A e. B \/ B e. A ) ) )
2322adantl 482 . . . 4 |- ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) -> ( A = B <-> -. ( A e. B \/ B e. A ) ) )
2423necon2abid 2836 . . 3 |- ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) -> ( ( A e. B \/ B e. A ) <-> A =/= B ) )
2518, 24mpbird 247 . 2 |- ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) -> ( A e. B \/ B e. A ) )
2611, 17, 25mpjaodan 827 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 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   ran crn 5115   Ord word 5722   suc csuc 5725   -->wf 5884   ` cfv 5888   omcom 7065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fv 5896  df-om 7066
This theorem is referenced by:  isf32lem7  9181
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