Theorem elqsi 4297

Description: Membership in a quotient set.

Assertion

Ref	Expression
elqsi	|- (B e. (A/.R) -> E.x(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 1484	. . . 4 |- (y = B -> (y = [x]R <-> B = [x]R))
2	1	anbi2d 618	. . 3 |- (y = B -> ((x e. A /\ y = [x]R) <-> (x e. A /\ B = [x]R)))
3	2	exbidv 1281	. 2 |- (y = B -> (E.x(x e. A /\ y = [x]R) <-> E.x(x e. A /\ B = [x]R)))
4		visset 1816	. . . 4 |- y e. V
5	4	elqs 4296	. . 3 |- (y e. (A/.R) <-> E.x(x e. A /\ y = [x]R))
6	5	biimp 151	. 2 |- (y e. (A/.R) -> E.x(x e. A /\ y = [x]R))
7	3, 6	vtoclga 1855	1 |- (B e. (A/.R) -> E.x(x e. A /\ B = [x]R))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  [cec 4265  /.cqs 4266

This theorem is referenced by:  0nelqs 4304  ectocl 4308  ecoptocl 4309

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-12 970  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rex 1653  df-v 1815  df-qs 4272

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