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Theorem pssdif 4309
 Description: A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
Assertion
Ref Expression
pssdif (𝐴𝐵 → (𝐵𝐴) ≠ ∅)

Proof of Theorem pssdif
StepHypRef Expression
1 df-pss 3938 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
2 pssdifn0 4308 . 2 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
31, 2sylbi 220 1 (𝐴𝐵 → (𝐵𝐴) ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ≠ wne 3014   ∖ cdif 3916   ⊆ wss 3919   ⊊ wpss 3920  ∅c0 4276 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277 This theorem is referenced by:  pssnel  4403  pgpfac1lem5  19201  fundmpss  33066  dfon2lem6  33090
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