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Metamath Proof Explorer

Theorem elimasn

Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011) (Proof shortened by BJ, 16-Oct-2024) TODO: replace existing usages by usages of elimasn1 , remove, and relabel elimasn1 to "elimasn".

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

Proof

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