Theorem supex2g

Description: Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Assertion	supex2g	$$⊢A\in C\to sup\left(B,A,R\right)\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		df-sup	$$⊢sup\left(B,A,R\right)=\bigcup \left\{x\in A|\left(\forall y\in B\neg xRy\wedge \forall y\in A\left(yRx\to \exists z\in ByRz\right)\right)\right\}$$
2		rabexg	$$⊢A\in C\to \left\{x\in A|\left(\forall y\in B\neg xRy\wedge \forall y\in A\left(yRx\to \exists z\in ByRz\right)\right)\right\}\in \mathrm{V}$$
3	2	uniexd	$$⊢A\in C\to \bigcup \left\{x\in A|\left(\forall y\in B\neg xRy\wedge \forall y\in A\left(yRx\to \exists z\in ByRz\right)\right)\right\}\in \mathrm{V}$$
4	1 3	eqeltrid	$$⊢A\in C\to sup\left(B,A,R\right)\in \mathrm{V}$$