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Theorem dfpss3 3726
 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 3725 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
2 eqss 3651 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32baib 964 . . . 4 (𝐴𝐵 → (𝐴 = 𝐵𝐵𝐴))
43notbid 307 . . 3 (𝐴𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵𝐴))
54pm5.32i 670 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
61, 5bitri 264 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 383   = wceq 1523   ⊆ wss 3607   ⊊ wpss 3608 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-ne 2824  df-in 3614  df-ss 3621  df-pss 3623 This theorem is referenced by:  pssirr  3740  pssn2lp  3741  ssnpss  3743  nsspssun  3890  npss0OLD  4048  pssdifcom1  4087  pssdifcom2  4088  php3  8187  fincssdom  9183  reclem2pr  9908  ressval3d  15984  islbs3  19203  chpsscon3  28490  chpssati  29350  fundmpss  31790  lpssat  34618  lssat  34621  dihglblem6  36946  pssnssi  39597  mbfpsssmf  41312
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