Theorem elimasn1

Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011) Use df-br and shorten proof. (Revised by BJ, 16-Oct-2024)

	Ref	Expression
Hypotheses	elimasn1.1	$$⊢B\in \mathrm{V}$$
	elimasn1.2	$$⊢C\in \mathrm{V}$$
Assertion	elimasn1	$$⊢C\in A\left[\left\{B\right\}\right]\leftrightarrow BAC$$

Proof

Step	Hyp	Ref	Expression
1		elimasn1.1	$$⊢B\in \mathrm{V}$$
2		elimasn1.2	$$⊢C\in \mathrm{V}$$
3		elimasng1	$$⊢\left(B\in \mathrm{V}\wedge C\in \mathrm{V}\right)\to \left(C\in A\left[\left\{B\right\}\right]\leftrightarrow BAC\right)$$
4	1 2 3	mp2an	$$⊢C\in A\left[\left\{B\right\}\right]\leftrightarrow BAC$$