Theorem ideqg 3282

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 3279	. . . . 5 |- Rel I
2	1	brrelexi 3214	. . . 4 |- (AIB -> A e. V)
3	2	adantl 390	. . 3 |- ((B e. C /\ AIB) -> A e. V)
4		elisset 1820	. . . 4 |- (B e. C -> B e. V)
5	4	adantr 391	. . 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 1537	. . . . 5 |- (A = B -> (A e. C <-> B e. C))
8	7	biimparc 421	. . . 4 |- ((B e. C /\ A = B) -> A e. C)
9		elisset 1820	. . . 4 |- (A e. C -> A e. V)
10	8, 9	syl 10	. . 3 |- ((B e. C /\ A = B) -> A e. V)
11	4	adantr 391	. . 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 1484	. . 3 |- (x = A -> (x = y <-> A = y))
14		eqeq2 1487	. . 3 |- (y = B -> (A = y <-> A = B))
15		df-id 2841	. . 3 |- I = {<.x, y>. | x = y}
16	13, 14, 15	brabg 2824	. 2 |- ((A e. V /\ B e. V) -> (AIB <-> A = B))
17	6, 12, 16	pm5.21nd 682	1 |- (B e. C -> (AIB <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814   class class class wbr 2624  Icid 2837

This theorem is referenced by:  ideq 3283  issetid 3286

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191

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