Theorem elrabi

Description: Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017)

	Ref	Expression
Assertion	elrabi	$$⊢A\in \left\{x\in V|\phi \right\}\to A\in V$$

Proof

Step	Hyp	Ref	Expression
1		clelab	$$⊢A\in \left\{x|\left(x\in V\wedge \phi \right)\right\}\leftrightarrow \exists x\left(x=A\wedge \left(x\in V\wedge \phi \right)\right)$$
2		eleq1	$$⊢x=A\to \left(x\in V\leftrightarrow A\in V\right)$$
3	2	anbi1d	$$⊢x=A\to \left(\left(x\in V\wedge \phi \right)\leftrightarrow \left(A\in V\wedge \phi \right)\right)$$
4	3	simprbda	$$⊢\left(x=A\wedge \left(x\in V\wedge \phi \right)\right)\to A\in V$$
5	4	exlimiv	$$⊢\exists x\left(x=A\wedge \left(x\in V\wedge \phi \right)\right)\to A\in V$$
6	1 5	sylbi	$$⊢A\in \left\{x|\left(x\in V\wedge \phi \right)\right\}\to A\in V$$
7		df-rab	$$⊢\left\{x\in V|\phi \right\}=\left\{x|\left(x\in V\wedge \phi \right)\right\}$$
8	6 7	eleq2s	$$⊢A\in \left\{x\in V|\phi \right\}\to A\in V$$