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Theorem npss0 3991
 Description: No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
Assertion
Ref Expression
npss0 ¬ 𝐴 ⊊ ∅

Proof of Theorem npss0
StepHypRef Expression
1 0ss 3949 . 2 ∅ ⊆ 𝐴
2 ssnpss 3693 . 2 (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅)
31, 2ax-mp 5 1 ¬ 𝐴 ⊊ ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ⊆ wss 3560   ⊊ wpss 3561  ∅c0 3896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-v 3193  df-dif 3563  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897 This theorem is referenced by:  pssnn  8123
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