Theorem fri

Description: A nonempty subset of an R -well-founded class has an R -minimal element (inference form). (Contributed by BJ, 16-Nov-2024) (Proof shortened by BJ, 19-Nov-2024)

	Ref	Expression
Assertion	fri	$$⊢\left(\left(B\in C\wedge R\mathrm{Fr}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to \exists x\in B\forall y\in B\neg yRx$$

Proof

Step	Hyp	Ref	Expression
1		simplr	$$⊢\left(\left(B\in C\wedge R\mathrm{Fr}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to R\mathrm{Fr}A$$
2		simprl	$$⊢\left(\left(B\in C\wedge R\mathrm{Fr}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to B\subseteq A$$
3		simpll	$$⊢\left(\left(B\in C\wedge R\mathrm{Fr}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to B\in C$$
4		simprr	$$⊢\left(\left(B\in C\wedge R\mathrm{Fr}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to B\ne \varnothing$$
5	1 2 3 4	frd	$$⊢\left(\left(B\in C\wedge R\mathrm{Fr}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to \exists x\in B\forall y\in B\neg yRx$$