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Metamath Proof Explorer

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 )