Theorem preqsn

Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008) (Revised by AV, 12-Jun-2022)

	Ref	Expression
Hypotheses	preqsn.1	⊢ 𝐴 ∈ V
	preqsn.2	⊢ 𝐵 ∈ V
Assertion	preqsn	⊢ ( { 𝐴 , 𝐵 } = { 𝐶 } ↔ ( 𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		preqsn.1	⊢ 𝐴 ∈ V
2		preqsn.2	⊢ 𝐵 ∈ V
3		id	⊢ ( 𝐴 ∈ V → 𝐴 ∈ V )
4	2	a1i	⊢ ( 𝐴 ∈ V → 𝐵 ∈ V )
5	3 4	preqsnd	⊢ ( 𝐴 ∈ V → ( { 𝐴 , 𝐵 } = { 𝐶 } ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) ) )
6	1 5	ax-mp	⊢ ( { 𝐴 , 𝐵 } = { 𝐶 } ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) )
7		eqeq2	⊢ ( 𝐵 = 𝐶 → ( 𝐴 = 𝐵 ↔ 𝐴 = 𝐶 ) )
8	7	pm5.32ri	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ) ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) )
9	6 8	bitr4i	⊢ ( { 𝐴 , 𝐵 } = { 𝐶 } ↔ ( 𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ) )