Theorem elqsi 3228

Description: Membership in a quotient set.

Assertion

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

Distinct variable group(s):   x,A   x,B   x,R

Proof of Theorem elqsi

Step	Hyp	Ref	Expression
1		cleq1 1107	. . . 4 |- (y = B -> (y = [x]R <-> B = [x]R))
2	1	anbi2d 468	. . 3 |- (y = B -> ((x e. A /\ y = [x]R) <-> (x e. A /\ B = [x]R)))
3	2	biexdv 936	. 2 |- (y = B -> (E.x(x e. A /\ y = [x]R) <-> E.x(x e. A /\ B = [x]R)))
4		visset 1350	. . . 4 |- y e. V
5	4	elqs 3227	. . 3 |- (y e. (A/.R) <-> E.x(x e. A /\ y = [x]R))
6	5	biimp 133	. 2 |- (y e. (A/.R) -> E.x(x e. A /\ y = [x]R))
7	3, 6	vtoclga 1387	1 |- (B e. (A/.R) -> E.x(x e. A /\ B = [x]R))

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  [cec 3198  /.cqs 3199

This theorem is referenced by:  0nelqs 3234  ectocl 3238  ecoptocl 3239

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-qs 3205

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