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Theorem dfpss3 4089
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 4088 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
2 eqss 3999 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32baib 535 . . . 4 (𝐴𝐵 → (𝐴 = 𝐵𝐵𝐴))
43notbid 318 . . 3 (𝐴𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵𝐴))
54pm5.32i 574 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
61, 5bitri 275 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1540  wss 3951  wpss 3952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-ne 2941  df-ss 3968  df-pss 3971
This theorem is referenced by:  pssirr  4103  pssn2lp  4104  ssnpss  4106  nsspssun  4268  pssdifcom1  4490  pssdifcom2  4491  php3  9249  php3OLD  9261  fincssdom  10363  reclem2pr  11088  ressval3d  17292  islbs3  21157  sltlpss  27945  chpsscon3  31522  chpssati  32382  fundmpss  35767  lpssat  39014  lssat  39017  dihglblem6  41342  pssnssi  45106  mbfpsssmf  46798
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