Theorem bnj1309

Description: Technical lemma for bnj60 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj1309.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
Assertion	bnj1309	$$⊢w\in B\to \forall xw\in B$$

Proof

Step	Hyp	Ref	Expression
1		bnj1309.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
2		hbra1	$$⊢\forall x\in dpred\left(x,A,R\right)\subseteq d\to \forall x\forall x\in dpred\left(x,A,R\right)\subseteq d$$
3	2	bnj1352	$$⊢\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\to \forall x\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)$$
4	3	hbab	$$⊢w\in \left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}\to \forall xw\in \left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
5	1 4	hbxfreq	$$⊢w\in B\to \forall xw\in B$$