Theorem eqssi 3209

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	|- A C_ B
eqssi.2	|- B C_ A

Assertion

Ref	Expression
eqssi	|- A = B

Proof of Theorem eqssi

Step	Hyp	Ref	Expression
1		eqssi.1	. 2 |- A C_ B
2		eqssi.2	. 2 |- B C_ A
3		eqss 3208	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
4	1, 2, 3	mpbir2an 945	1 |- A = B

Syntax hints:   = wceq 1373   C_ wss 3166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  inv1  3497  unv  3498  undifabs  3537  intab  3914  intid  4269  find  4648  limom  4663  dmv  4895  0ima  5043  rnxpid  5118  dftpos4  6351  axaddf  7983  axmulf  7984  dfuzi  9485  unirnioo  10097  0bits  12303  4sqlem19  12765  txuni2  14761  dvef  15232  reeff1o  15278