Theorem rabsnel

Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018)

	Ref	Expression
Hypothesis	rabsnel.1	$$⊢B\in \mathrm{V}$$
Assertion	rabsnel	$$⊢\left\{x\in A|\phi \right\}=\left\{B\right\}\to B\in A$$

Proof

Step	Hyp	Ref	Expression
1		rabsnel.1	$$⊢B\in \mathrm{V}$$
2	1	snid	$$⊢B\in \left\{B\right\}$$
3		eleq2	$$⊢\left\{x\in A|\phi \right\}=\left\{B\right\}\to \left(B\in \left\{x\in A|\phi \right\}\leftrightarrow B\in \left\{B\right\}\right)$$
4	2 3	mpbiri	$$⊢\left\{x\in A|\phi \right\}=\left\{B\right\}\to B\in \left\{x\in A|\phi \right\}$$
5		elrabi	$$⊢B\in \left\{x\in A|\phi \right\}\to B\in A$$
6	4 5	syl	$$⊢\left\{x\in A|\phi \right\}=\left\{B\right\}\to B\in A$$