Theorem vss

Description: Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	vss	⊢ ( V ⊆ 𝐴 ↔ 𝐴 = V )

Proof

Step	Hyp	Ref	Expression
1		ssv	⊢ 𝐴 ⊆ V
2	1	biantrur	⊢ ( V ⊆ 𝐴 ↔ ( 𝐴 ⊆ V ∧ V ⊆ 𝐴 ) )
3		eqss	⊢ ( 𝐴 = V ↔ ( 𝐴 ⊆ V ∧ V ⊆ 𝐴 ) )
4	2 3	bitr4i	⊢ ( V ⊆ 𝐴 ↔ 𝐴 = V )