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Theorem dfpss3 3355
 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 ↔ (A B ¬ B A))

Proof of Theorem dfpss3
StepHypRef Expression
1 dfpss2 3354 . 2 (AB ↔ (A B ¬ A = B))
2 eqss 3287 . . . . 5 (A = B ↔ (A B B A))
32baib 871 . . . 4 (A B → (A = BB A))
43notbid 285 . . 3 (A B → (¬ A = B ↔ ¬ B A))
54pm5.32i 618 . 2 ((A B ¬ A = B) ↔ (A B ¬ B A))
61, 5bitri 240 1 (AB ↔ (A B ¬ B A))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   = wceq 1642   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261 This theorem is referenced by:  pssirr  3369  pssn2lp  3370  ssnpss  3372  nsspssun  3488  npss0  3589  pssdifcom1  3635  pssdifcom2  3636  dfpss4  3888
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