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 = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
Assertion	bnj1309	|- ( w e. B -> A. x w e. B )

Proof

Step	Hyp	Ref	Expression
1		bnj1309.1	|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2		hbra1	|- ( A. x e. d _pred ( x , A , R ) C_ d -> A. x A. x e. d _pred ( x , A , R ) C_ d )
3	2	bnj1352	|- ( ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) -> A. x ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) )
4	3	hbab	|- ( w e. { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) } -> A. x w e. { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) } )
5	1 4	hbxfreq	|- ( w e. B -> A. x w e. B )