Theorem elqsi 6553

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 6551	. 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 1343   e. wcel 2136   E.wrex 2445   [cec 6499   /.cqs 6500

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-qs 6507

This theorem is referenced by:  ectocld  6567  ecoptocl  6588  eroveu  6592  dmaddpqlem  7318  nqpi  7319  nq0nn  7383