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Theorem pssdif 4109
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 3748 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
2 pssdifn0 4108 . 2 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
31, 2sylbi 208 1 (𝐴𝐵 → (𝐵𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wne 2937  cdif 3729  wss 3732  wpss 3733  c0 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-v 3352  df-dif 3735  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080
This theorem is referenced by:  pssnel  4199  pgpfac1lem5  18745  fundmpss  32109  dfon2lem6  32136
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