Theorem pssn0 39189

Description: A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)

Assertion

Ref	Expression
pssn0	⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅)

Proof of Theorem pssn0

Step	Hyp	Ref	Expression
1		npss0 4390	. . 3 ⊢ ¬ 𝐴 ⊊ ∅
2		psseq2 4058	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ⊊ 𝐵 ↔ 𝐴 ⊊ ∅))
3	1, 2	mtbiri 329	. 2 ⊢ (𝐵 = ∅ → ¬ 𝐴 ⊊ 𝐵)
4	3	necon2ai 3044	1 ⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1536   ≠ wne 3015   ⊊ wpss 3930  ∅c0 4284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-ne 3016  df-dif 3932  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285

This theorem is referenced by:  xppss12  39191