Theorem ideqg 3276

Description: For sets, the identity relation is the same as equality.

Assertion

Ref	Expression
ideqg	|- (B e. C -> (AIB <-> A = B))

Proof of Theorem ideqg

Step	Hyp	Ref	Expression
1		reli 3273	. . . . 5 |- Rel I
2	1	brrelexi 3208	. . . 4 |- (AIB -> A e. V)
3	2	adantl 388	. . 3 |- ((B e. C /\ AIB) -> A e. V)
4		elisset 1817	. . . 4 |- (B e. C -> B e. V)
5	4	adantr 389	. . 3 |- ((B e. C /\ AIB) -> B e. V)
6	3, 5	jca 288	. 2 |- ((B e. C /\ AIB) -> (A e. V /\ B e. V))
7		eleq1 1534	. . . . 5 |- (A = B -> (A e. C <-> B e. C))
8	7	biimparc 419	. . . 4 |- ((B e. C /\ A = B) -> A e. C)
9		elisset 1817	. . . 4 |- (A e. C -> A e. V)
10	8, 9	syl 10	. . 3 |- ((B e. C /\ A = B) -> A e. V)
11	4	adantr 389	. . 3 |- ((B e. C /\ A = B) -> B e. V)
12	10, 11	jca 288	. 2 |- ((B e. C /\ A = B) -> (A e. V /\ B e. V))
13		eqeq1 1481	. . 3 |- (x = A -> (x = y <-> A = y))
14		eqeq2 1484	. . 3 |- (y = B -> (A = y <-> A = B))
15		df-id 2835	. . 3 |- I = {<.x, y>. | x = y}
16	13, 14, 15	brabg 2818	. 2 |- ((A e. V /\ B e. V) -> (AIB <-> A = B))
17	6, 12, 16	pm5.21nd 680	1 |- (B e. C -> (AIB <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   class class class wbr 2619  Icid 2831

This theorem is referenced by:  ideq 3277  issetid 3280

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185

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