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Theorem dfpss3 4061
 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 4060 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
2 eqss 3980 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32baib 538 . . . 4 (𝐴𝐵 → (𝐴 = 𝐵𝐵𝐴))
43notbid 320 . . 3 (𝐴𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵𝐴))
54pm5.32i 577 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
61, 5bitri 277 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   = wceq 1530   ⊆ wss 3934   ⊊ wpss 3935 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-ne 3015  df-in 3941  df-ss 3950  df-pss 3952 This theorem is referenced by:  pssirr  4075  pssn2lp  4076  ssnpss  4078  nsspssun  4232  pssdifcom1  4433  pssdifcom2  4434  php3  8695  fincssdom  9737  reclem2pr  10462  ressval3d  16553  islbs3  19919  chpsscon3  29272  chpssati  30132  fundmpss  32997  lpssat  36136  lssat  36139  dihglblem6  38463  pssnssi  41352  mbfpsssmf  43044
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