Theorem tz6.26

Description: All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of TakeutiZaring p. 31. (Contributed by Scott Fenton, 29-Jan-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Assertion	tz6.26	$$⊢\left(\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to \exists y\in BPred\left(R,B,y\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		wereu2	$$⊢\left(\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to \exists !y\in B\forall x\in B\neg xRy$$
2		reurex	$$⊢\exists !y\in B\forall x\in B\neg xRy\to \exists y\in B\forall x\in B\neg xRy$$
3	1 2	syl	$$⊢\left(\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to \exists y\in B\forall x\in B\neg xRy$$
4		rabeq0	$$⊢\left\{x\in B|xRy\right\}=\varnothing \leftrightarrow \forall x\in B\neg xRy$$
5		dfrab3	$$⊢\left\{x\in B|xRy\right\}=B\cap \left\{x|xRy\right\}$$
6		vex	$$⊢y\in \mathrm{V}$$
7	6	dfpred2	$$⊢Pred\left(R,B,y\right)=B\cap \left\{x|xRy\right\}$$
8	5 7	eqtr4i	$$⊢\left\{x\in B|xRy\right\}=Pred\left(R,B,y\right)$$
9	8	eqeq1i	$$⊢\left\{x\in B|xRy\right\}=\varnothing \leftrightarrow Pred\left(R,B,y\right)=\varnothing$$
10	4 9	bitr3i	$$⊢\forall x\in B\neg xRy\leftrightarrow Pred\left(R,B,y\right)=\varnothing$$
11	10	rexbii	$$⊢\exists y\in B\forall x\in B\neg xRy\leftrightarrow \exists y\in BPred\left(R,B,y\right)=\varnothing$$
12	3 11	sylib	$$⊢\left(\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to \exists y\in BPred\left(R,B,y\right)=\varnothing$$