Theorem frc

Description: Property of well-founded relation (one direction of definition using class variables). (Contributed by NM, 17-Feb-2004) (Revised by Mario Carneiro, 19-Nov-2014)

	Ref	Expression
Hypothesis	frc.1	$$⊢B\in \mathrm{V}$$
Assertion	frc	$$⊢\left(R\mathrm{Fr}A\wedge B\subseteq A\wedge B\ne \varnothing \right)\to \exists x\in B\left\{y\in B|yRx\right\}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		frc.1	$$⊢B\in \mathrm{V}$$
2		fri	$$⊢\left(\left(B\in \mathrm{V}\wedge R\mathrm{Fr}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to \exists x\in B\forall z\in B\neg zRx$$
3	1 2	mpanl1	$$⊢\left(R\mathrm{Fr}A\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to \exists x\in B\forall z\in B\neg zRx$$
4	3	3impb	$$⊢\left(R\mathrm{Fr}A\wedge B\subseteq A\wedge B\ne \varnothing \right)\to \exists x\in B\forall z\in B\neg zRx$$
5		breq1	$$⊢y=z\to \left(yRx\leftrightarrow zRx\right)$$
6	5	rabeq0w	$$⊢\left\{y\in B|yRx\right\}=\varnothing \leftrightarrow \forall z\in B\neg zRx$$
7	6	rexbii	$$⊢\exists x\in B\left\{y\in B|yRx\right\}=\varnothing \leftrightarrow \exists x\in B\forall z\in B\neg zRx$$
8	4 7	sylibr	$$⊢\left(R\mathrm{Fr}A\wedge B\subseteq A\wedge B\ne \varnothing \right)\to \exists x\in B\left\{y\in B|yRx\right\}=\varnothing$$