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Metamath Proof Explorer

Theorem sssseq

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)

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

Proof

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