Theorem bnj168 30798

Description: First-order logic and set theory. Revised to remove dependence on ax-reg 8497. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj168.1	|- D = ( om \ { (/) } )

Assertion

Ref	Expression
bnj168	|- ( ( n =/= 1o /\ n e. D ) -> E. m e. D n = suc m )

Distinct variable group:   m, n

Allowed substitution hints:   D( m, n)

Proof of Theorem bnj168

Step	Hyp	Ref	Expression
1		bnj168.1	. . . . . . . . . 10 |- D = ( om \ { (/) } )
2	1	bnj158 30797	. . . . . . . . 9 |- ( n e. D -> E. m e. om n = suc m )
3	2	anim2i 593	. . . . . . . 8 |- ( ( n =/= 1o /\ n e. D ) -> ( n =/= 1o /\ E. m e. om n = suc m ) )
4		r19.42v 3092	. . . . . . . 8 |- ( E. m e. om ( n =/= 1o /\ n = suc m ) <-> ( n =/= 1o /\ E. m e. om n = suc m ) )
5	3, 4	sylibr 224	. . . . . . 7 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n =/= 1o /\ n = suc m ) )
6		neeq1 2856	. . . . . . . . . . 11 |- ( n = suc m -> ( n =/= 1o <-> suc m =/= 1o ) )
7	6	biimpac 503	. . . . . . . . . 10 |- ( ( n =/= 1o /\ n = suc m ) -> suc m =/= 1o )
8		df-1o 7560	. . . . . . . . . . . . 13 |- 1o = suc (/)
9	8	eqeq2i 2634	. . . . . . . . . . . 12 |- ( suc m = 1o <-> suc m = suc (/) )
10		nnon 7071	. . . . . . . . . . . . 13 |- ( m e. om -> m e. On )
11		0elon 5778	. . . . . . . . . . . . 13 |- (/) e. On
12		suc11 5831	. . . . . . . . . . . . 13 |- ( ( m e. On /\ (/) e. On ) -> ( suc m = suc (/) <-> m = (/) ) )
13	10, 11, 12	sylancl 694	. . . . . . . . . . . 12 |- ( m e. om -> ( suc m = suc (/) <-> m = (/) ) )
14	9, 13	syl5rbb 273	. . . . . . . . . . 11 |- ( m e. om -> ( m = (/) <-> suc m = 1o ) )
15	14	necon3bid 2838	. . . . . . . . . 10 |- ( m e. om -> ( m =/= (/) <-> suc m =/= 1o ) )
16	7, 15	syl5ibr 236	. . . . . . . . 9 |- ( m e. om -> ( ( n =/= 1o /\ n = suc m ) -> m =/= (/) ) )
17	16	ancld 576	. . . . . . . 8 |- ( m e. om -> ( ( n =/= 1o /\ n = suc m ) -> ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) ) )
18	17	reximia 3009	. . . . . . 7 |- ( E. m e. om ( n =/= 1o /\ n = suc m ) -> E. m e. om ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) )
19	5, 18	syl 17	. . . . . 6 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) )
20		anass 681	. . . . . . 7 |- ( ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) <-> ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) )
21	20	rexbii 3041	. . . . . 6 |- ( E. m e. om ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) <-> E. m e. om ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) )
22	19, 21	sylib 208	. . . . 5 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) )
23		simpr 477	. . . . 5 |- ( ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) -> ( n = suc m /\ m =/= (/) ) )
24	22, 23	bnj31 30785	. . . 4 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n = suc m /\ m =/= (/) ) )
25		df-rex 2918	. . . 4 |- ( E. m e. om ( n = suc m /\ m =/= (/) ) <-> E. m ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) )
26	24, 25	sylib 208	. . 3 |- ( ( n =/= 1o /\ n e. D ) -> E. m ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) )
27		simpr 477	. . . . . . 7 |- ( ( n = suc m /\ m =/= (/) ) -> m =/= (/) )
28	27	anim2i 593	. . . . . 6 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> ( m e. om /\ m =/= (/) ) )
29	1	eleq2i 2693	. . . . . . 7 |- ( m e. D <-> m e. ( om \ { (/) } ) )
30		eldifsn 4317	. . . . . . 7 |- ( m e. ( om \ { (/) } ) <-> ( m e. om /\ m =/= (/) ) )
31	29, 30	bitr2i 265	. . . . . 6 |- ( ( m e. om /\ m =/= (/) ) <-> m e. D )
32	28, 31	sylib 208	. . . . 5 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> m e. D )
33		simprl 794	. . . . 5 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> n = suc m )
34	32, 33	jca 554	. . . 4 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> ( m e. D /\ n = suc m ) )
35	34	eximi 1762	. . 3 |- ( E. m ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> E. m ( m e. D /\ n = suc m ) )
36	26, 35	syl 17	. 2 |- ( ( n =/= 1o /\ n e. D ) -> E. m ( m e. D /\ n = suc m ) )
37		df-rex 2918	. 2 |- ( E. m e. D n = suc m <-> E. m ( m e. D /\ n = suc m ) )
38	36, 37	sylibr 224	1 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. D n = suc m )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   (/)c0 3915   {csn 4177   Oncon0 5723   suc csuc 5725   omcom 7065   1oc1o 7553

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-1o 7560

This theorem is referenced by:  bnj600  30989