Theorem npss0 4399

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

Step	Hyp	Ref	Expression
1		0ss 4351	. 2 ⊢ ∅ ⊆ 𝐴
2		ssnpss 4058	. 2 ⊢ (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐴 ⊊ ∅

Syntax hints:  ¬ wn 3   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-dif 3905  df-ss 3919  df-pss 3922  df-nul 4284

This theorem is referenced by:  pssnn  9130  pssn0  42806