Theorem eqssi 3241

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 ⊆ 𝐵
eqssi.2	⊢ 𝐵 ⊆ 𝐴

Assertion

Ref	Expression
eqssi	⊢ 𝐴 = 𝐵

Proof of Theorem eqssi

Step	Hyp	Ref	Expression
1		eqssi.1	. 2 ⊢ 𝐴 ⊆ 𝐵
2		eqssi.2	. 2 ⊢ 𝐵 ⊆ 𝐴
3		eqss 3240	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
4	1, 2, 3	mpbir2an 948	1 ⊢ 𝐴 = 𝐵

Syntax hints:   = wceq 1395   ⊆ wss 3198

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3204  df-ss 3211

This theorem is referenced by:  inv1  3529  unv  3530  undifabs  3569  intab  3955  intid  4314  find  4695  limom  4710  dmv  4945  0ima  5094  rnxpid  5169  dftpos4  6424  axaddf  8078  axmulf  8079  dfuzi  9580  unirnioo  10198  0bits  12510  4sqlem19  12972  txuni2  14970  dvef  15441  reeff1o  15487