Theorem ideqg 4114

Description: For sets, the identity relation is the same as equality. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

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

Proof of Theorem ideqg

Step	Hyp	Ref	Expression
1		reli 4105	. . . . 5 |- Rel _I
2	1	brrelexi 4029	. . . 4 |- (A _I B -> A e. _V)
3	2	adantl 424	. . 3 |- ((B e. C /\ A _I B) -> A e. _V)
4		simpl 346	. . 3 |- ((B e. C /\ A _I B) -> B e. C)
5	3, 4	jca 310	. 2 |- ((B e. C /\ A _I B) -> (A e. _V /\ B e. C))
6		eleq1 1957	. . . . 5 |- (A = B -> (A e. C <-> B e. C))
7	6	biimparc 463	. . . 4 |- ((B e. C /\ A = B) -> A e. C)
8		elisset 2299	. . . 4 |- (A e. C -> A e. _V)
9	7, 8	syl 12	. . 3 |- ((B e. C /\ A = B) -> A e. _V)
10		simpl 346	. . 3 |- ((B e. C /\ A = B) -> B e. C)
11	9, 10	jca 310	. 2 |- ((B e. C /\ A = B) -> (A e. _V /\ B e. C))
12		eqeq1 1890	. . 3 |- (x = A -> (x = y <-> A = y))
13		eqeq2 1893	. . 3 |- (y = B -> (A = y <-> A = B))
14		df-id 3586	. . 3 |- _I = {<.x, y>. | x = y}
15	12, 13, 14	brabg 3568	. 2 |- ((A e. _V /\ B e. C) -> (A _I B <-> A = B))
16	5, 11, 15	pm5.21nd 744	1 |- (B e. C -> (A _I B <-> A = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338   _I cid 3582

This theorem is referenced by:  ideq 4116  ididg 4117  issetidOLD 4122  restidsing 14391  pltval 16781

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-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-id 3586  df-xp 4000  df-rel 4001

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