Theorem elqsi 6360

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 6358	. 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 1290   e. wcel 1439   E.wrex 2361   [cec 6306   /.cqs 6307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2624  df-qs 6314

This theorem is referenced by:  ectocld  6374  ecoptocl  6395  eroveu  6399  dmaddpqlem  6999  nqpi  7000  nq0nn  7064