Theorem npss0 4455

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 4407	. 2 ⊢ ∅ ⊆ 𝐴
2		ssnpss 4117	. 2 ⊢ (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐴 ⊊ ∅

Syntax hints:  ¬ wn 3   ⊆ wss 3964   ⊊ wpss 3965  ∅c0 4340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-dif 3967  df-ss 3981  df-pss 3984  df-nul 4341

This theorem is referenced by:  pssnn  9213  pssn0  42257