Theorem eqssi 3108

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 3107	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
4	1, 2, 3	mpbir2an 926	1 ⊢ 𝐴 = 𝐵

Syntax hints:   = wceq 1331   ⊆ wss 3066

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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  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-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079

This theorem is referenced by:  inv1  3394  unv  3395  undifabs  3434  intab  3795  intid  4141  find  4508  limom  4522  dmv  4750  0ima  4894  rnxpid  4968  dftpos4  6153  axaddf  7669  axmulf  7670  dfuzi  9154  unirnioo  9749  txuni2  12414  dvef  12845