Theorem pssn0 43026

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 4367	. . 3 ⊢ ¬ 𝐴 ⊊ ∅
2		psseq2 4044	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ⊊ 𝐵 ↔ 𝐴 ⊊ ∅))
3	1, 2	mtbiri 330	. 2 ⊢ (𝐵 = ∅ → ¬ 𝐴 ⊊ 𝐵)
4	3	necon2ai 2986	1 ⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ≠ wne 2957   ⊊ wpss 3905  ∅c0 4285

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-dif 3907  df-ss 3921  df-pss 3924  df-nul 4286

This theorem is used by:  xppss12  43028