Theorem bnj1309 31090

Description: Technical lemma for bnj60 31130. 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.)

Hypothesis

Ref	Expression
bnj1309.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }

Assertion

Ref	Expression
bnj1309	|- ( w e. B -> A. x w e. B )

Distinct variable groups:   x, A   x, d   x, w

Allowed substitution hints:   A( w, d)   B( x, w, d)   R( x, w, d)

Proof of Theorem bnj1309

Step	Hyp	Ref	Expression
1		bnj1309.1	. 2 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
2		hbra1 2942	. . . 4 |- ( 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 30898	. . 3 |- ( ( 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 2613	. 2 |- ( 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 2730	1 |- ( w e. B -> A. x w e. B )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   C_ wss 3574   predc-bnj14 30754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917

This theorem is referenced by:  bnj1311  31092  bnj1373  31098  bnj1498  31129  bnj1525  31137  bnj1523  31139