Theorem pssn0 40202

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 4379	. . 3 ⊢ ¬ 𝐴 ⊊ ∅
2		psseq2 4023	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ⊊ 𝐵 ↔ 𝐴 ⊊ ∅))
3	1, 2	mtbiri 327	. 2 ⊢ (𝐵 = ∅ → ¬ 𝐴 ⊊ 𝐵)
4	3	necon2ai 2973	1 ⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1539   ≠ wne 2943   ⊊ wpss 3888  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257

This theorem is referenced by:  xppss12  40204