Theorem elqsi 6181

Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

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		elqsg 6179	. 2 |- ( B e. ( A /. R ) -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )
2	1	ibi 174	1 |- ( B e. ( A /. R ) -> E. x e. A B = [ x ] R )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   E.wrex 2349   [cec 6127   /.cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-qs 6135

This theorem is referenced by:  ectocld  6195  ecoptocl  6216  eroveu  6220  dmaddpqlem  6567  nqpi  6568  nq0nn  6632