Theorem bnj168 29326

Description: First-order logic and set theory. Revised to remove dependence on ax-reg 8107. (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 29325	. . . . . . . . 9 |- ( n e. D -> E. m e. om n = suc m )
3	2	anim2i 571	. . . . . . . 8 |- ( ( n =/= 1o /\ n e. D ) -> ( n =/= 1o /\ E. m e. om n = suc m ) )
4		r19.42v 2990	. . . . . . . 8 |- ( E. m e. om ( n =/= 1o /\ n = suc m ) <-> ( n =/= 1o /\ E. m e. om n = suc m ) )
5	3, 4	sylibr 215	. . . . . . 7 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n =/= 1o /\ n = suc m ) )
6		neeq1 2712	. . . . . . . . . . 11 |- ( n = suc m -> ( n =/= 1o <-> suc m =/= 1o ) )
7	6	biimpac 488	. . . . . . . . . 10 |- ( ( n =/= 1o /\ n = suc m ) -> suc m =/= 1o )
8		df-1o 7190	. . . . . . . . . . . . 13 |- 1o = suc (/)
9	8	eqeq2i 2447	. . . . . . . . . . . 12 |- ( suc m = 1o <-> suc m = suc (/) )
10		nnon 6712	. . . . . . . . . . . . 13 |- ( m e. om -> m e. On )
11		0elon 5495	. . . . . . . . . . . . 13 |- (/) e. On
12		suc11 5545	. . . . . . . . . . . . 13 |- ( ( m e. On /\ (/) e. On ) -> ( suc m = suc (/) <-> m = (/) ) )
13	10, 11, 12	sylancl 666	. . . . . . . . . . . 12 |- ( m e. om -> ( suc m = suc (/) <-> m = (/) ) )
14	9, 13	syl5rbb 261	. . . . . . . . . . 11 |- ( m e. om -> ( m = (/) <-> suc m = 1o ) )
15	14	necon3bid 2689	. . . . . . . . . 10 |- ( m e. om -> ( m =/= (/) <-> suc m =/= 1o ) )
16	7, 15	syl5ibr 224	. . . . . . . . 9 |- ( m e. om -> ( ( n =/= 1o /\ n = suc m ) -> m =/= (/) ) )
17	16	ancld 555	. . . . . . . 8 |- ( m e. om -> ( ( n =/= 1o /\ n = suc m ) -> ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) ) )
18	17	reximia 2898	. . . . . . 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 653	. . . . . . 7 |- ( ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) <-> ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) )
21	20	rexbii 2934	. . . . . 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 199	. . . . 5 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) )
23		simpr 462	. . . . 5 |- ( ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) -> ( n = suc m /\ m =/= (/) ) )
24	22, 23	bnj31 29313	. . . 4 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n = suc m /\ m =/= (/) ) )
25		df-rex 2788	. . . 4 |- ( E. m e. om ( n = suc m /\ m =/= (/) ) <-> E. m ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) )
26	24, 25	sylib 199	. . 3 |- ( ( n =/= 1o /\ n e. D ) -> E. m ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) )
27		simpr 462	. . . . . . 7 |- ( ( n = suc m /\ m =/= (/) ) -> m =/= (/) )
28	27	anim2i 571	. . . . . 6 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> ( m e. om /\ m =/= (/) ) )
29	1	eleq2i 2507	. . . . . . 7 |- ( m e. D <-> m e. ( om \ { (/) } ) )
30		eldifsn 4128	. . . . . . 7 |- ( m e. ( om \ { (/) } ) <-> ( m e. om /\ m =/= (/) ) )
31	29, 30	bitr2i 253	. . . . . 6 |- ( ( m e. om /\ m =/= (/) ) <-> m e. D )
32	28, 31	sylib 199	. . . . 5 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> m e. D )
33		simprl 762	. . . . 5 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> n = suc m )
34	32, 33	jca 534	. . . 4 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> ( m e. D /\ n = suc m ) )
35	34	eximi 1703	. . 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 2788	. 2 |- ( E. m e. D n = suc m <-> E. m ( m e. D /\ n = suc m ) )
38	36, 37	sylibr 215	1 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. D n = suc m )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1659   e. wcel 1870   =/= wne 2625   E.wrex 2783   \ cdif 3439   (/)c0 3767   {csn 4002   Oncon0 5442   suc csuc 5444   omcom 6706   1oc1o 7183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-om 6707  df-1o 7190

This theorem is referenced by:  bnj600  29518