Theorem bnj168 32054

Description: First-order logic and set theory. Revised to remove dependence on ax-reg 7919. (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 32053	. . . . . . . . 9 |- ( n e. D -> E. m e. om n = suc m )
3	2	anim2i 569	. . . . . . . 8 |- ( ( n =/= 1o /\ n e. D ) -> ( n =/= 1o /\ E. m e. om n = suc m ) )
4		r19.42v 2981	. . . . . . . 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 2733	. . . . . . . . . . 11 |- ( n = suc m -> ( n =/= 1o <-> suc m =/= 1o ) )
7	6	biimpac 486	. . . . . . . . . 10 |- ( ( n =/= 1o /\ n = suc m ) -> suc m =/= 1o )
8		df-1o 7031	. . . . . . . . . . . . 13 |- 1o = suc (/)
9	8	eqeq2i 2472	. . . . . . . . . . . 12 |- ( suc m = 1o <-> suc m = suc (/) )
10		nnon 6593	. . . . . . . . . . . . 13 |- ( m e. om -> m e. On )
11		0elon 4881	. . . . . . . . . . . . 13 |- (/) e. On
12		suc11 4931	. . . . . . . . . . . . 13 |- ( ( m e. On /\ (/) e. On ) -> ( suc m = suc (/) <-> m = (/) ) )
13	10, 11, 12	sylancl 662	. . . . . . . . . . . 12 |- ( m e. om -> ( suc m = suc (/) <-> m = (/) ) )
14	9, 13	syl5rbb 258	. . . . . . . . . . 11 |- ( m e. om -> ( m = (/) <-> suc m = 1o ) )
15	14	necon3bid 2710	. . . . . . . . . 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 553	. . . . . . . 8 |- ( m e. om -> ( ( n =/= 1o /\ n = suc m ) -> ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) ) )
18	17	reximia 2927	. . . . . . 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 649	. . . . . . 7 |- ( ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) <-> ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) )
21	20	rexbii 2862	. . . . . 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 461	. . . . 5 |- ( ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) -> ( n = suc m /\ m =/= (/) ) )
24	22, 23	bnj31 32041	. . . 4 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n = suc m /\ m =/= (/) ) )
25		df-rex 2805	. . . 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 461	. . . . . . 7 |- ( ( n = suc m /\ m =/= (/) ) -> m =/= (/) )
28	27	anim2i 569	. . . . . 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 4109	. . . . . . 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 755	. . . . 5 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> n = suc m )
34	32, 33	jca 532	. . . 4 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> ( m e. D /\ n = suc m ) )
35	34	eximi 1626	. . 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 2805	. 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 369   = wceq 1370   E.wex 1587   e. wcel 1758   =/= wne 2648   E.wrex 2800   \ cdif 3434   (/)c0 3746   {csn 3986   Oncon0 4828   suc csuc 4830   omcom 6587   1oc1o 7024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-tr 4495  df-eprel 4741  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-om 6588  df-1o 7031

This theorem is referenced by:  bnj600  32245