Theorem elqs 6157

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

Hypothesis

Ref	Expression
elqs.1	|- B e. _V

Assertion

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

Distinct variable groups:   x, A   x, B   x, R

Proof of Theorem elqs

Step	Hyp	Ref	Expression
1		elqs.1	. 2 |- B e. _V
2		elqsg 6156	. 2 |- ( B e. _V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )
3	1, 2	ax-mp 7	1 |- ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R )

Syntax hints:   <-> wb 98   = wceq 1243   e. wcel 1393   E.wrex 2307   _Vcvv 2557   [cec 6104   /.cqs 6105

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-qs 6112

This theorem is referenced by:  qsss  6165  qsid  6171  erovlem  6198  nqnq0  6539