Theorem bnj970 31017

Description: Technical lemma for bnj69 31078. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj970.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj970.10	|- D = ( om \ { (/) } )

Assertion

Ref	Expression
bnj970	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> p e. D )

Proof of Theorem bnj970

Step	Hyp	Ref	Expression
1		bnj970.3	. . . . 5 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
2	1	bnj1232 30874	. . . 4 |- ( ch -> n e. D )
3	2	3ad2ant1 1082	. . 3 |- ( ( ch /\ n = suc m /\ p = suc n ) -> n e. D )
4	3	adantl 482	. 2 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> n e. D )
5		simpr3 1069	. 2 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> p = suc n )
6		bnj970.10	. . . . 5 |- D = ( om \ { (/) } )
7	6	bnj923 30838	. . . 4 |- ( n e. D -> n e. om )
8		peano2 7086	. . . . 5 |- ( n e. om -> suc n e. om )
9		eleq1 2689	. . . . 5 |- ( p = suc n -> ( p e. om <-> suc n e. om ) )
10		bianir 1009	. . . . 5 |- ( ( suc n e. om /\ ( p e. om <-> suc n e. om ) ) -> p e. om )
11	8, 9, 10	syl2an 494	. . . 4 |- ( ( n e. om /\ p = suc n ) -> p e. om )
12	7, 11	sylan 488	. . 3 |- ( ( n e. D /\ p = suc n ) -> p e. om )
13		df-suc 5729	. . . . . 6 |- suc n = ( n u. { n } )
14	13	eqeq2i 2634	. . . . 5 |- ( p = suc n <-> p = ( n u. { n } ) )
15		ssun2 3777	. . . . . . 7 |- { n } C_ ( n u. { n } )
16		vex 3203	. . . . . . . 8 |- n e. _V
17	16	snnz 4309	. . . . . . 7 |- { n } =/= (/)
18		ssn0 3976	. . . . . . 7 |- ( ( { n } C_ ( n u. { n } ) /\ { n } =/= (/) ) -> ( n u. { n } ) =/= (/) )
19	15, 17, 18	mp2an 708	. . . . . 6 |- ( n u. { n } ) =/= (/)
20		neeq1 2856	. . . . . 6 |- ( p = ( n u. { n } ) -> ( p =/= (/) <-> ( n u. { n } ) =/= (/) ) )
21	19, 20	mpbiri 248	. . . . 5 |- ( p = ( n u. { n } ) -> p =/= (/) )
22	14, 21	sylbi 207	. . . 4 |- ( p = suc n -> p =/= (/) )
23	22	adantl 482	. . 3 |- ( ( n e. D /\ p = suc n ) -> p =/= (/) )
24	6	eleq2i 2693	. . . 4 |- ( p e. D <-> p e. ( om \ { (/) } ) )
25		eldifsn 4317	. . . 4 |- ( p e. ( om \ { (/) } ) <-> ( p e. om /\ p =/= (/) ) )
26	24, 25	bitri 264	. . 3 |- ( p e. D <-> ( p e. om /\ p =/= (/) ) )
27	12, 23, 26	sylanbrc 698	. 2 |- ( ( n e. D /\ p = suc n ) -> p e. D )
28	4, 5, 27	syl2anc 693	1 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> p e. D )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   suc csuc 5725   Fn wfn 5883   omcom 7065   /\ w-bnj17 30752   FrSe w-bnj15 30758

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-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj910  31018