ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elqsi Unicode version

Theorem elqsi 6474
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
StepHypRef Expression
1 elqsg 6472 . 2  |-  ( B  e.  ( A /. R )  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
) )
21ibi 175 1  |-  ( B  e.  ( A /. R )  ->  E. x  e.  A  B  =  [ x ] R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   E.wrex 2415   [cec 6420   /.cqs 6421
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-qs 6428
This theorem is referenced by:  ectocld  6488  ecoptocl  6509  eroveu  6513  dmaddpqlem  7178  nqpi  7179  nq0nn  7243
  Copyright terms: Public domain W3C validator