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Theorem dfpss3 3055
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 3054 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
2 eqss 2985 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32baib 837 . . . 4 (𝐴𝐵 → (𝐴 = 𝐵𝐵𝐴))
43notbid 600 . . 3 (𝐴𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵𝐴))
54pm5.32i 435 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
61, 5bitri 177 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101  wb 102   = wceq 1257  wss 2942  wpss 2943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-11 1411  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-ne 2219  df-in 2949  df-ss 2956  df-pss 2958
This theorem is referenced by:  pssirr  3069  pssn2lp  3070  ssnpss  3072  nsspssun  3195  npss0  3291
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