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Theorem dfpss3 3003
 Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
dfpss3 (AB ↔ (AB ¬ BA))

Proof of Theorem dfpss3
StepHypRef Expression
1 dfpss2 3002 . 2 (AB ↔ (AB ¬ A = B))
2 eqss 2933 . . . . 5 (A = B ↔ (AB BA))
32baib 816 . . . 4 (AB → (A = BBA))
43notbid 579 . . 3 (AB → (¬ A = B ↔ ¬ BA))
54pm5.32i 430 . 2 ((AB ¬ A = B) ↔ (AB ¬ BA))
61, 5bitri 173 1 (AB ↔ (AB ¬ BA))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   = wceq 1226   ⊆ wss 2890   ⊊ wpss 2891 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-in1 532  ax-in2 533  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-ne 2184  df-in 2897  df-ss 2904  df-pss 2906 This theorem is referenced by:  pssirr  3017  pssn2lp  3018  ssnpss  3020  nsspssun  3143  npss0  3239
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