Theorem sseq0b 4356

Description: The only subclass of the empty class is itself. (Contributed by NM, 7-Mar-2007.) Strengthen sseq0 4357 to a biconditional. (Revised by BJ, 19-Jul-2026.)

Assertion

Ref	Expression
sseq0b	⊢ (𝐴 = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 = ∅))

Proof of Theorem sseq0b

Step	Hyp	Ref	Expression
1		sseq2 3960	. 2 ⊢ (𝐴 = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ ∅))
2		ss0b 4354	. 2 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
3	1, 2	bitrdi 290	1 ⊢ (𝐴 = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 = ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ⊆ wss 3902  ∅c0 4282

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-dif 3905  df-ss 3919  df-nul 4283

This theorem is used by:  sseq0  4357