Theorem moabex

Description: "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996)

	Ref	Expression
Assertion	moabex	$$⊢{\exists }^{*}x\phi \to \left\{x|\phi \right\}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		df-mo	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$
2		abss	$$⊢\left\{x|\phi \right\}\subseteq \left\{y\right\}\leftrightarrow \forall x\left(\phi \to x\in \left\{y\right\}\right)$$
3		velsn	$$⊢x\in \left\{y\right\}\leftrightarrow x=y$$
4	3	imbi2i	$$⊢\left(\phi \to x\in \left\{y\right\}\right)\leftrightarrow \left(\phi \to x=y\right)$$
5	4	albii	$$⊢\forall x\left(\phi \to x\in \left\{y\right\}\right)\leftrightarrow \forall x\left(\phi \to x=y\right)$$
6	2 5	bitri	$$⊢\left\{x|\phi \right\}\subseteq \left\{y\right\}\leftrightarrow \forall x\left(\phi \to x=y\right)$$
7		vsnex	$$⊢\left\{y\right\}\in \mathrm{V}$$
8	7	ssex	$$⊢\left\{x|\phi \right\}\subseteq \left\{y\right\}\to \left\{x|\phi \right\}\in \mathrm{V}$$
9	6 8	sylbir	$$⊢\forall x\left(\phi \to x=y\right)\to \left\{x|\phi \right\}\in \mathrm{V}$$
10	9	exlimiv	$$⊢\exists y\forall x\left(\phi \to x=y\right)\to \left\{x|\phi \right\}\in \mathrm{V}$$
11	1 10	sylbi	$$⊢{\exists }^{*}x\phi \to \left\{x|\phi \right\}\in \mathrm{V}$$