Theorem bnj112 12457

Description: First-order logic and set theory.

Hypotheses

Ref	Expression
bnj112.1	|- D = (om \ {(/)})
bnj112.2	|- (ta <-> A.m e. D (m _E n -> [m / n]th))

Assertion

Ref	Expression
bnj112	|- (n = 1o -> ta)

Distinct variable group:   m,n

Proof of Theorem bnj112

Step	Hyp	Ref	Expression
1		ax-17 1317	. . . 4 |- (n = 1o -> A.m n = 1o)
2		breq2 3342	. . . . . 6 |- (n = 1o -> (m _E n <-> m _E 1o))
3		visset 2295	. . . . . . . 8 |- m e. _V
4		bnj105 12451	. . . . . . . 8 |- 1o e. _V
5	3, 4	epelc 3584	. . . . . . 7 |- (m _E 1o <-> m e. 1o)
6		el1o 5191	. . . . . . 7 |- (m e. 1o <-> m = (/))
7	5, 6	bitr2i 191	. . . . . 6 |- (m = (/) <-> m _E 1o)
8	2, 7	syl6bbr 597	. . . . 5 |- (n = 1o -> (m _E n <-> m = (/)))
9		elndif 2732	. . . . . 6 |- (m e. {(/)} -> -. m e. (om \ {(/)}))
10		elsn 3058	. . . . . 6 |- (m e. {(/)} <-> m = (/))
11		bnj112.1	. . . . . . . . 9 |- D = (om \ {(/)})
12	11	eleq2i 1961	. . . . . . . 8 |- (m e. D <-> m e. (om \ {(/)}))
13	12	notbii 204	. . . . . . 7 |- (-. m e. D <-> -. m e. (om \ {(/)}))
14	13	bicomi 189	. . . . . 6 |- (-. m e. (om \ {(/)}) <-> -. m e. D)
15	9, 10, 14	3imtr3i 235	. . . . 5 |- (m = (/) -> -. m e. D)
16	8, 15	syl6bi 231	. . . 4 |- (n = 1o -> (m _E n -> -. m e. D))
17	1, 16	19.21ai 1345	. . 3 |- (n = 1o -> A.m(m _E n -> -. m e. D))
18		iman 256	. . . . . . 7 |- ((m _E n -> -. m e. D) <-> -. (m _E n /\ -. -. m e. D))
19		notnot 178	. . . . . . . . . 10 |- (m e. D <-> -. -. m e. D)
20	19	bicomi 189	. . . . . . . . 9 |- (-. -. m e. D <-> m e. D)
21	20	anbi2i 538	. . . . . . . 8 |- ((m _E n /\ -. -. m e. D) <-> (m _E n /\ m e. D))
22	21	notbii 204	. . . . . . 7 |- (-. (m _E n /\ -. -. m e. D) <-> -. (m _E n /\ m e. D))
23		ancom 482	. . . . . . . 8 |- ((m _E n /\ m e. D) <-> (m e. D /\ m _E n))
24	23	notbii 204	. . . . . . 7 |- (-. (m _E n /\ m e. D) <-> -. (m e. D /\ m _E n))
25	18, 22, 24	3bitri 194	. . . . . 6 |- ((m _E n -> -. m e. D) <-> -. (m e. D /\ m _E n))
26	25	biimpi 168	. . . . 5 |- ((m _E n -> -. m e. D) -> -. (m e. D /\ m _E n))
27	26	pm2.21d 94	. . . 4 |- ((m _E n -> -. m e. D) -> ((m e. D /\ m _E n) -> [m / n]th))
28	27	alimi 1338	. . 3 |- (A.m(m _E n -> -. m e. D) -> A.m((m e. D /\ m _E n) -> [m / n]th))
29	17, 28	syl 12	. 2 |- (n = 1o -> A.m((m e. D /\ m _E n) -> [m / n]th))
30		bnj112.2	. . 3 |- (ta <-> A.m e. D (m _E n -> [m / n]th))
31		df-ral 2109	. . 3 |- (A.m e. D (m _E n -> [m / n]th) <-> A.m(m e. D -> (m _E n -> [m / n]th)))
32		impexp 374	. . . . 5 |- (((m e. D /\ m _E n) -> [m / n]th) <-> (m e. D -> (m _E n -> [m / n]th)))
33	32	bicomi 189	. . . 4 |- ((m e. D -> (m _E n -> [m / n]th)) <-> ((m e. D /\ m _E n) -> [m / n]th))
34	33	albii 1346	. . 3 |- (A.m(m e. D -> (m _E n -> [m / n]th)) <-> A.m((m e. D /\ m _E n) -> [m / n]th))
35	30, 31, 34	3bitri 194	. 2 |- (ta <-> A.m((m e. D /\ m _E n) -> [m / n]th))
36	29, 35	sylibr 217	1 |- (n = 1o -> ta)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105   \ cdif 2590  (/)c0 2875  {csn 3044   class class class wbr 3338   _E cep 3581  omcom 3949  1oc1o 5172

This theorem is referenced by:  bnj116 12460

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-eprel 3583  df-suc 3663  df-1o 5177

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