Theorem isf32lem3 9177

Description: Lemma for isfin3-2 9189. Being a chain, difference sets are disjoint (one case). (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
isf32lem3	|- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> ( ( ( 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 isf32lem3

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 3732	. . . 4 |- ( a e. ( ( F ` A ) \ ( F ` suc A ) ) -> a e. ( F ` A ) )
2		simpll 790	. . . . . 6 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> A e. om )
3		peano2 7086	. . . . . . 7 |- ( B e. om -> suc B e. om )
4	3	ad2antlr 763	. . . . . 6 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> suc B e. om )
5		nnord 7073	. . . . . . . 8 |- ( A e. om -> Ord A )
6	5	ad2antrr 762	. . . . . . 7 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> Ord A )
7		simprl 794	. . . . . . 7 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> B e. A )
8		ordsucss 7018	. . . . . . 7 |- ( Ord A -> ( B e. A -> suc B C_ A ) )
9	6, 7, 8	sylc 65	. . . . . 6 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> suc B C_ A )
10		simprr 796	. . . . . 6 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> ph )
11		isf32lem.a	. . . . . . 7 |- ( ph -> F : om --> ~P G )
12		isf32lem.b	. . . . . . 7 |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )
13		isf32lem.c	. . . . . . 7 |- ( ph -> -. |^| ran F e. ran F )
14	11, 12, 13	isf32lem1 9175	. . . . . 6 |- ( ( ( A e. om /\ suc B e. om ) /\ ( suc B C_ A /\ ph ) ) -> ( F ` A ) C_ ( F ` suc B ) )
15	2, 4, 9, 10, 14	syl22anc 1327	. . . . 5 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> ( F ` A ) C_ ( F ` suc B ) )
16	15	sseld 3602	. . . 4 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> ( a e. ( F ` A ) -> a e. ( F ` suc B ) ) )
17		elndif 3734	. . . 4 |- ( a e. ( F ` suc B ) -> -. a e. ( ( F ` B ) \ ( F ` suc B ) ) )
18	1, 16, 17	syl56 36	. . 3 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> ( a e. ( ( F ` A ) \ ( F ` suc A ) ) -> -. a e. ( ( F ` B ) \ ( F ` suc B ) ) ) )
19	18	ralrimiv 2965	. 2 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> A. a e. ( ( F ` A ) \ ( F ` suc A ) ) -. a e. ( ( F ` B ) \ ( F ` suc B ) ) )
20		disj 4017	. 2 |- ( ( ( ( F ` A ) \ ( F ` suc A ) ) i^i ( ( F ` B ) \ ( F ` suc B ) ) ) = (/) <-> A. a e. ( ( F ` A ) \ ( F ` suc A ) ) -. a e. ( ( F ` B ) \ ( F ` suc B ) ) )
21	19, 20	sylibr 224	1 |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> ( ( ( F ` A ) \ ( F ` suc A ) ) i^i ( ( F ` B ) \ ( F ` suc B ) ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   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:  isf32lem4  9178