Theorem elqsi 5349

Description: Membership in a quotient set.

Assertion

Ref	Expression
elqsi	|- (B e. (A/.R) -> E.x e. A B = [x]R)

Distinct variable groups:   x,A   x,B   x,R

Proof of Theorem elqsi

Step	Hyp	Ref	Expression
1		eqeq1 1890	. . 3 |- (y = B -> (y = [x]R <-> B = [x]R))
2	1	rexbidv 2124	. 2 |- (y = B -> (E.x e. A y = [x]R <-> E.x e. A B = [x]R))
3		visset 2295	. . . 4 |- y e. _V
4	3	elqs 5348	. . 3 |- (y e. (A/.R) <-> E.x e. A y = [x]R)
5	4	biimpi 168	. 2 |- (y e. (A/.R) -> E.x e. A y = [x]R)
6	2, 5	vtoclga 2352	1 |- (B e. (A/.R) -> E.x e. A B = [x]R)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  E.wrex 2106  [cec 5316  /.cqs 5317

This theorem is referenced by:  0nelqs 5357  ectocl 5361  ecoptocl 5362  erdisj3 16266  0nelqs2 16269

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294  df-qs 5323

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