Theorem npss0 4360

Description: No set is a proper subset of the empty set. Dual of nvpss 4362. (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

Step	Hyp	Ref	Expression
1		0ss 4349	. 2 ⊢ ∅ ⊆ 𝐴
2		ssnpss 4034	. 2 ⊢ (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐴 ⊊ ∅

This proof depends on syntax axioms:  ¬ wn 3   ⊆ wss 3895   ⊊ wpss 4013  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ss 3896  df-pss 4014  df-dif 4047  df-nul 4279

This theorem is used by:  pssnn  9188  pssn0  43375