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Theorem pssdif 4369
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 3971 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
2 pssdifn0 4368 . 2 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
31, 2sylbi 217 1 (𝐴𝐵 → (𝐵𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wne 2940  cdif 3948  wss 3951  wpss 3952  c0 4333
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-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-ss 3968  df-pss 3971  df-nul 4334
This theorem is referenced by:  pssnel  4471  pgpfac1lem5  20099  fundmpss  35767  dfon2lem6  35789
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