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Theorem pssdif 3943
 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 3588 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
2 pssdifn0 3942 . 2 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
31, 2sylbi 207 1 (𝐴𝐵 → (𝐵𝐴) ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ≠ wne 2793   ∖ cdif 3569   ⊆ wss 3572   ⊊ wpss 3573  ∅c0 3913 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-v 3200  df-dif 3575  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914 This theorem is referenced by:  pssnel  4037  pgpfac1lem5  18472  fundmpss  31650  dfon2lem6  31677
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