Theorem elqsi 6489

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 6487	. 2 |- ( B e. ( A /. R ) -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )
2	1	ibi 175	1 |- ( B e. ( A /. R ) -> E. x e. A B = [ x ] R )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   E.wrex 2418   [cec 6435   /.cqs 6436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-qs 6443

This theorem is referenced by:  ectocld  6503  ecoptocl  6524  eroveu  6528  dmaddpqlem  7209  nqpi  7210  nq0nn  7274