Theorem bnj168 12496

Description: First-order logic and set theory.

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

Proof of Theorem bnj168

Step	Hyp	Ref	Expression
1		bnj168.1	. . . . . . . . . 10 |- D = (om \ {(/)})
2	1	bnj158 12483	. . . . . . . . 9 |- (n e. D -> E.m e. om n = suc m)
3	2	anim2i 362	. . . . . . . 8 |- ((n =/= 1o /\ n e. D) -> (n =/= 1o /\ E.m e. om n = suc m))
4		r19.42v 2237	. . . . . . . 8 |- (E.m e. om (n =/= 1o /\ n = suc m) <-> (n =/= 1o /\ E.m e. om n = suc m))
5	3, 4	sylibr 217	. . . . . . 7 |- ((n =/= 1o /\ n e. D) -> E.m e. om (n =/= 1o /\ n = suc m))
6		neeq1 2024	. . . . . . . . . 10 |- (n = suc m -> (n =/= 1o <-> suc m =/= 1o))
7	6	biimpac 462	. . . . . . . . 9 |- ((n =/= 1o /\ n = suc m) -> suc m =/= 1o)
8		df-1o 5177	. . . . . . . . . . . 12 |- 1o = suc (/)
9	8	eqeq2i 1894	. . . . . . . . . . 11 |- (suc m = 1o <-> suc m = suc (/))
10		suc11reg 5710	. . . . . . . . . . 11 |- (suc m = suc (/) <-> m = (/))
11	9, 10	bitr2i 191	. . . . . . . . . 10 |- (m = (/) <-> suc m = 1o)
12	11	necon3bii 2032	. . . . . . . . 9 |- (m =/= (/) <-> suc m =/= 1o)
13	7, 12	sylibr 217	. . . . . . . 8 |- ((n =/= 1o /\ n = suc m) -> m =/= (/))
14	13	bnj165 12493	. . . . . . 7 |- (E.m e. om (n =/= 1o /\ n = suc m) -> E.m e. om ((n =/= 1o /\ n = suc m) /\ m =/= (/)))
15	5, 14	syl 12	. . . . . 6 |- ((n =/= 1o /\ n e. D) -> E.m e. om ((n =/= 1o /\ n = suc m) /\ m =/= (/)))
16		anass 487	. . . . . . 7 |- (((n =/= 1o /\ n = suc m) /\ m =/= (/)) <-> (n =/= 1o /\ (n = suc m /\ m =/= (/))))
17	16	rexbii 2128	. . . . . 6 |- (E.m e. om ((n =/= 1o /\ n = suc m) /\ m =/= (/)) <-> E.m e. om (n =/= 1o /\ (n = suc m /\ m =/= (/))))
18	15, 17	sylib 215	. . . . 5 |- ((n =/= 1o /\ n e. D) -> E.m e. om (n =/= 1o /\ (n = suc m /\ m =/= (/))))
19		simpr 350	. . . . 5 |- ((n =/= 1o /\ (n = suc m /\ m =/= (/))) -> (n = suc m /\ m =/= (/)))
20	18, 19	bnj31 12400	. . . 4 |- ((n =/= 1o /\ n e. D) -> E.m e. om (n = suc m /\ m =/= (/)))
21		df-rex 2110	. . . 4 |- (E.m e. om (n = suc m /\ m =/= (/)) <-> E.m(m e. om /\ (n = suc m /\ m =/= (/))))
22	20, 21	sylib 215	. . 3 |- ((n =/= 1o /\ n e. D) -> E.m(m e. om /\ (n = suc m /\ m =/= (/))))
23		simpr 350	. . . . . . 7 |- ((n = suc m /\ m =/= (/)) -> m =/= (/))
24	23	anim2i 362	. . . . . 6 |- ((m e. om /\ (n = suc m /\ m =/= (/))) -> (m e. om /\ m =/= (/)))
25	1	eleq2i 1961	. . . . . . 7 |- (m e. D <-> m e. (om \ {(/)}))
26		eldifsn 3123	. . . . . . 7 |- (m e. (om \ {(/)}) <-> (m e. om /\ m =/= (/)))
27	25, 26	bitr2i 191	. . . . . 6 |- ((m e. om /\ m =/= (/)) <-> m e. D)
28	24, 27	sylib 215	. . . . 5 |- ((m e. om /\ (n = suc m /\ m =/= (/))) -> m e. D)
29		simprl 450	. . . . 5 |- ((m e. om /\ (n = suc m /\ m =/= (/))) -> n = suc m)
30	28, 29	jca 310	. . . 4 |- ((m e. om /\ (n = suc m /\ m =/= (/))) -> (m e. D /\ n = suc m))
31	30	eximi 1387	. . 3 |- (E.m(m e. om /\ (n = suc m /\ m =/= (/))) -> E.m(m e. D /\ n = suc m))
32	22, 31	syl 12	. 2 |- ((n =/= 1o /\ n e. D) -> E.m(m e. D /\ n = suc m))
33		df-rex 2110	. 2 |- (E.m e. D n = suc m <-> E.m(m e. D /\ n = suc m))
34	32, 33	sylibr 217	1 |- ((n =/= 1o /\ n e. D) -> E.m e. D n = suc m)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  E.wrex 2106   \ cdif 2590  (/)c0 2875  {csn 3044  suc csuc 3659  omcom 3949  1oc1o 5172

This theorem is referenced by:  bnj542 12536

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-1o 5177

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