Theorem sneqr

Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of TakeutiZaring p. 15. (Contributed by NM, 27-Aug-1993)

	Ref	Expression
Hypothesis	sneqr.1	⊢ 𝐴 ∈ V
Assertion	sneqr	⊢ ( { 𝐴 } = { 𝐵 } → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sneqr.1	⊢ 𝐴 ∈ V
2		sneqrg	⊢ ( 𝐴 ∈ V → ( { 𝐴 } = { 𝐵 } → 𝐴 = 𝐵 ) )
3	1 2	ax-mp	⊢ ( { 𝐴 } = { 𝐵 } → 𝐴 = 𝐵 )