ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqssi Structured version   GIF version

Theorem eqssi 2934
Description: Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
Hypotheses
Ref Expression
eqssi.1 AB
eqssi.2 BA
Assertion
Ref Expression
eqssi A = B

Proof of Theorem eqssi
StepHypRef Expression
1 eqssi.1 . 2 AB
2 eqssi.2 . 2 BA
3 eqss 2933 . 2 (A = B ↔ (AB BA))
41, 2, 3mpbir2an 835 1 A = B
Colors of variables: wff set class
Syntax hints:   = wceq 1226  wss 2890
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-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-in 2897  df-ss 2904
This theorem is referenced by:  inv1  3226  unv  3227  undifabs  3273  intab  3614  intid  3930  find  4245  limom  4259  dmv  4474  0ima  4608  rnxpid  4678  dftpos4  5796  tfrlemi14  5865
  Copyright terms: Public domain W3C validator