Theorem pssn0 42207

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 4419	. . 3 ⊢ ¬ 𝐴 ⊊ ∅
2		psseq2 4062	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ⊊ 𝐵 ↔ 𝐴 ⊊ ∅))
3	1, 2	mtbiri 327	. 2 ⊢ (𝐵 = ∅ → ¬ 𝐴 ⊊ 𝐵)
4	3	necon2ai 2956	1 ⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1540   ≠ wne 2927   ⊊ wpss 3923  ∅c0 4304

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2928  df-dif 3925  df-ss 3939  df-pss 3942  df-nul 4305

This theorem is referenced by:  xppss12  42209