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	$$⊢B\in \mathrm{V}$$
	elimasn.2	$$⊢C\in \mathrm{V}$$
Assertion	elimasn	$$⊢C\in A\left[\left\{B\right\}\right]\leftrightarrow ⟨B,C⟩\in A$$

Proof

Step	Hyp	Ref	Expression
1		elimasn.1	$$⊢B\in \mathrm{V}$$
2		elimasn.2	$$⊢C\in \mathrm{V}$$
3		elimasng	$$⊢\left(B\in \mathrm{V}\wedge C\in \mathrm{V}\right)\to \left(C\in A\left[\left\{B\right\}\right]\leftrightarrow ⟨B,C⟩\in A\right)$$
4	1 2 3	mp2an	$$⊢C\in A\left[\left\{B\right\}\right]\leftrightarrow ⟨B,C⟩\in A$$