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Theorem pssn0 38035
Description: A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)
Assertion
Ref Expression
pssn0 (𝐴𝐵𝐵 ≠ ∅)

Proof of Theorem pssn0
StepHypRef Expression
1 npss0 4210 . . 3 ¬ 𝐴 ⊊ ∅
2 psseq2 3892 . . 3 (𝐵 = ∅ → (𝐴𝐵𝐴 ⊊ ∅))
31, 2mtbiri 319 . 2 (𝐵 = ∅ → ¬ 𝐴𝐵)
43necon2ai 3000 1 (𝐴𝐵𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wne 2971  wpss 3770  c0 4115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-v 3387  df-dif 3772  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116
This theorem is referenced by:  xppss12  38037
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