Theorem sssseq 3965

Description: If a class is a subclass of another class, then the classes are equal if and only if the other class is a subclass of the first class. (Contributed by AV, 23-Dec-2020.)

Assertion

Ref	Expression
sssseq	⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem sssseq

Step	Hyp	Ref	Expression
1		eqss 3962	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
2	1	rbaibr 538	1 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-in 3920  df-ss 3930

This theorem is referenced by:  vdiscusgrb  28541