Theorem sup00

Description: The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020)

	Ref	Expression
Assertion	sup00	$$⊢sup\left(B,\varnothing ,R\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		df-sup	$$⊢sup\left(B,\varnothing ,R\right)=\bigcup \left\{x\in \varnothing |\left(\forall y\in B\neg xRy\wedge \forall y\in \varnothing \left(yRx\to \exists z\in ByRz\right)\right)\right\}$$
2		rab0	$$⊢\left\{x\in \varnothing |\left(\forall y\in B\neg xRy\wedge \forall y\in \varnothing \left(yRx\to \exists z\in ByRz\right)\right)\right\}=\varnothing$$
3	2	unieqi	$$⊢\bigcup \left\{x\in \varnothing |\left(\forall y\in B\neg xRy\wedge \forall y\in \varnothing \left(yRx\to \exists z\in ByRz\right)\right)\right\}=\bigcup \varnothing$$
4		uni0	$$⊢\bigcup \varnothing =\varnothing$$
5	1 3 4	3eqtri	$$⊢sup\left(B,\varnothing ,R\right)=\varnothing$$