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Theorem eqssi 3288
 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
StepHypRef Expression
1 eqssi.1 . 2 A B
2 eqssi.2 . 2 B A
3 eqss 3287 . 2 (A = B ↔ (A B B A))
41, 2, 3mpbir2an 886 1 A = B
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  inv1  3577  unv  3578  intab  3956  evenoddnnnul  4514  dmv  4920  0ima  5014  clos10  5887  cenc  6181
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