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Theorem pssdif 4359
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 3960 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
2 pssdifn0 4358 . 2 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
31, 2sylbi 216 1 (𝐴𝐵 → (𝐵𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wne 2932  cdif 3938  wss 3941  wpss 3942  c0 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316
This theorem is referenced by:  pssnel  4463  pgpfac1lem5  19997  fundmpss  35261  dfon2lem6  35283
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