Theorem bnj168 34186

Description: First-order logic and set theory. Revised to remove dependence on ax-reg 8010. (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 34185	. . . . . . . . 9 |- ( n e. D -> E. m e. om n = suc m )
3	2	anim2i 567	. . . . . . . 8 |- ( ( n =/= 1o /\ n e. D ) -> ( n =/= 1o /\ E. m e. om n = suc m ) )
4		r19.42v 3009	. . . . . . . 8 |- ( E. m e. om ( n =/= 1o /\ n = suc m ) <-> ( n =/= 1o /\ E. m e. om n = suc m ) )
5	3, 4	sylibr 212	. . . . . . 7 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n =/= 1o /\ n = suc m ) )
6		neeq1 2735	. . . . . . . . . . 11 |- ( n = suc m -> ( n =/= 1o <-> suc m =/= 1o ) )
7	6	biimpac 484	. . . . . . . . . 10 |- ( ( n =/= 1o /\ n = suc m ) -> suc m =/= 1o )
8		df-1o 7122	. . . . . . . . . . . . 13 |- 1o = suc (/)
9	8	eqeq2i 2472	. . . . . . . . . . . 12 |- ( suc m = 1o <-> suc m = suc (/) )
10		nnon 6679	. . . . . . . . . . . . 13 |- ( m e. om -> m e. On )
11		0elon 4920	. . . . . . . . . . . . 13 |- (/) e. On
12		suc11 4970	. . . . . . . . . . . . 13 |- ( ( m e. On /\ (/) e. On ) -> ( suc m = suc (/) <-> m = (/) ) )
13	10, 11, 12	sylancl 660	. . . . . . . . . . . 12 |- ( m e. om -> ( suc m = suc (/) <-> m = (/) ) )
14	9, 13	syl5rbb 258	. . . . . . . . . . 11 |- ( m e. om -> ( m = (/) <-> suc m = 1o ) )
15	14	necon3bid 2712	. . . . . . . . . 10 |- ( m e. om -> ( m =/= (/) <-> suc m =/= 1o ) )
16	7, 15	syl5ibr 221	. . . . . . . . 9 |- ( m e. om -> ( ( n =/= 1o /\ n = suc m ) -> m =/= (/) ) )
17	16	ancld 551	. . . . . . . 8 |- ( m e. om -> ( ( n =/= 1o /\ n = suc m ) -> ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) ) )
18	17	reximia 2920	. . . . . . 7 |- ( E. m e. om ( n =/= 1o /\ n = suc m ) -> E. m e. om ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) )
19	5, 18	syl 16	. . . . . 6 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) )
20		anass 647	. . . . . . 7 |- ( ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) <-> ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) )
21	20	rexbii 2956	. . . . . 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 196	. . . . 5 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) )
23		simpr 459	. . . . 5 |- ( ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) -> ( n = suc m /\ m =/= (/) ) )
24	22, 23	bnj31 34173	. . . 4 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n = suc m /\ m =/= (/) ) )
25		df-rex 2810	. . . 4 |- ( E. m e. om ( n = suc m /\ m =/= (/) ) <-> E. m ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) )
26	24, 25	sylib 196	. . 3 |- ( ( n =/= 1o /\ n e. D ) -> E. m ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) )
27		simpr 459	. . . . . . 7 |- ( ( n = suc m /\ m =/= (/) ) -> m =/= (/) )
28	27	anim2i 567	. . . . . 6 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> ( m e. om /\ m =/= (/) ) )
29	1	eleq2i 2532	. . . . . . 7 |- ( m e. D <-> m e. ( om \ { (/) } ) )
30		eldifsn 4141	. . . . . . 7 |- ( m e. ( om \ { (/) } ) <-> ( m e. om /\ m =/= (/) ) )
31	29, 30	bitr2i 250	. . . . . 6 |- ( ( m e. om /\ m =/= (/) ) <-> m e. D )
32	28, 31	sylib 196	. . . . 5 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> m e. D )
33		simprl 754	. . . . 5 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> n = suc m )
34	32, 33	jca 530	. . . 4 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> ( m e. D /\ n = suc m ) )
35	34	eximi 1661	. . 3 |- ( E. m ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> E. m ( m e. D /\ n = suc m ) )
36	26, 35	syl 16	. 2 |- ( ( n =/= 1o /\ n e. D ) -> E. m ( m e. D /\ n = suc m ) )
37		df-rex 2810	. 2 |- ( E. m e. D n = suc m <-> E. m ( m e. D /\ n = suc m ) )
38	36, 37	sylibr 212	1 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. D n = suc m )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   E.wex 1617   e. wcel 1823   =/= wne 2649   E.wrex 2805   \ cdif 3458   (/)c0 3783   {csn 4016   Oncon0 4867   suc csuc 4869   omcom 6673   1oc1o 7115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-om 6674  df-1o 7122

This theorem is referenced by:  bnj600  34378