Theorem npss0 4430

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 4382	. 2 ⊢ ∅ ⊆ 𝐴
2		ssnpss 4088	. 2 ⊢ (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐴 ⊊ ∅

Syntax hints:  ¬ wn 3   ⊆ wss 3933   ⊊ wpss 3934  ∅c0 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-dif 3936  df-ss 3950  df-pss 3953  df-nul 4316

This theorem is referenced by:  pssnn  9191  pssn0  42207