Theorem bnj168 28803

Description: First-order logic and set theory. Revised to remove dependence on ax-reg 7516. (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 28802	. . . . . . . . 9 |- ( n e. D -> E. m e. om n = suc m )
3	2	anim2i 553	. . . . . . . 8 |- ( ( n =/= 1o /\ n e. D ) -> ( n =/= 1o /\ E. m e. om n = suc m ) )
4		r19.42v 2822	. . . . . . . 8 |- ( E. m e. om ( n =/= 1o /\ n = suc m ) <-> ( n =/= 1o /\ E. m e. om n = suc m ) )
5	3, 4	sylibr 204	. . . . . . 7 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n =/= 1o /\ n = suc m ) )
6		neeq1 2575	. . . . . . . . . . 11 |- ( n = suc m -> ( n =/= 1o <-> suc m =/= 1o ) )
7	6	biimpac 473	. . . . . . . . . 10 |- ( ( n =/= 1o /\ n = suc m ) -> suc m =/= 1o )
8		df-1o 6683	. . . . . . . . . . . . 13 |- 1o = suc (/)
9	8	eqeq2i 2414	. . . . . . . . . . . 12 |- ( suc m = 1o <-> suc m = suc (/) )
10		nnon 4810	. . . . . . . . . . . . 13 |- ( m e. om -> m e. On )
11		0elon 4594	. . . . . . . . . . . . 13 |- (/) e. On
12		suc11 4644	. . . . . . . . . . . . 13 |- ( ( m e. On /\ (/) e. On ) -> ( suc m = suc (/) <-> m = (/) ) )
13	10, 11, 12	sylancl 644	. . . . . . . . . . . 12 |- ( m e. om -> ( suc m = suc (/) <-> m = (/) ) )
14	9, 13	syl5rbb 250	. . . . . . . . . . 11 |- ( m e. om -> ( m = (/) <-> suc m = 1o ) )
15	14	necon3bid 2602	. . . . . . . . . 10 |- ( m e. om -> ( m =/= (/) <-> suc m =/= 1o ) )
16	7, 15	syl5ibr 213	. . . . . . . . 9 |- ( m e. om -> ( ( n =/= 1o /\ n = suc m ) -> m =/= (/) ) )
17	16	ancld 537	. . . . . . . 8 |- ( m e. om -> ( ( n =/= 1o /\ n = suc m ) -> ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) ) )
18	17	reximia 2771	. . . . . . 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 631	. . . . . . 7 |- ( ( ( n =/= 1o /\ n = suc m ) /\ m =/= (/) ) <-> ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) )
21	20	rexbii 2691	. . . . . 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 189	. . . . 5 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) )
23		simpr 448	. . . . 5 |- ( ( n =/= 1o /\ ( n = suc m /\ m =/= (/) ) ) -> ( n = suc m /\ m =/= (/) ) )
24	22, 23	bnj31 28790	. . . 4 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. om ( n = suc m /\ m =/= (/) ) )
25		df-rex 2672	. . . 4 |- ( E. m e. om ( n = suc m /\ m =/= (/) ) <-> E. m ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) )
26	24, 25	sylib 189	. . 3 |- ( ( n =/= 1o /\ n e. D ) -> E. m ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) )
27		simpr 448	. . . . . . 7 |- ( ( n = suc m /\ m =/= (/) ) -> m =/= (/) )
28	27	anim2i 553	. . . . . 6 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> ( m e. om /\ m =/= (/) ) )
29	1	eleq2i 2468	. . . . . . 7 |- ( m e. D <-> m e. ( om \ { (/) } ) )
30		eldifsn 3887	. . . . . . 7 |- ( m e. ( om \ { (/) } ) <-> ( m e. om /\ m =/= (/) ) )
31	29, 30	bitr2i 242	. . . . . 6 |- ( ( m e. om /\ m =/= (/) ) <-> m e. D )
32	28, 31	sylib 189	. . . . 5 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> m e. D )
33		simprl 733	. . . . 5 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> n = suc m )
34	32, 33	jca 519	. . . 4 |- ( ( m e. om /\ ( n = suc m /\ m =/= (/) ) ) -> ( m e. D /\ n = suc m ) )
35	34	eximi 1582	. . 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 2672	. 2 |- ( E. m e. D n = suc m <-> E. m ( m e. D /\ n = suc m ) )
38	36, 37	sylibr 204	1 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. D n = suc m )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   E.wex 1547   = wceq 1649   e. wcel 1721   =/= wne 2567   E.wrex 2667   \ cdif 3277   (/)c0 3588   {csn 3774   Oncon0 4541   suc csuc 4543   omcom 4804   1oc1o 6676

This theorem is referenced by:  bnj600  28996

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-1o 6683