Theorem pssn0 43282

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 4360	. . 3 ⊢ ¬ 𝐴 ⊊ ∅
2		psseq2 4018	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ⊊ 𝐵 ↔ 𝐴 ⊊ ∅))
3	1, 2	mtbiri 330	. 2 ⊢ (𝐵 = ∅ → ¬ 𝐴 ⊊ 𝐵)
4	3	necon2ai 2984	1 ⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ≠ wne 2955   ⊊ wpss 4013  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ss 3896  df-pss 4014  df-dif 4047  df-nul 4279

This theorem is used by:  xppss12  43284