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Metamath Proof Explorer

Theorem opeqsn

Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008) (Revised by AV, 15-Jul-2022) (Avoid depending on this detail.)

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

Proof

Step	Hyp	Ref	Expression
1		opeqsn.1	⊢ 𝐴 ∈ V
2		opeqsn.2	⊢ 𝐵 ∈ V
3		opeqsng	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ⟨ 𝐴 , 𝐵 ⟩ = { 𝐶 } ↔ ( 𝐴 = 𝐵 ∧ 𝐶 = { 𝐴 } ) ) )
4	1 2 3	mp2an	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = { 𝐶 } ↔ ( 𝐴 = 𝐵 ∧ 𝐶 = { 𝐴 } ) )