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Theorem bnj1309 29368
Description: Technical lemma for bnj60 29408. 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
StepHypRef Expression
1 bnj1309.1 . 2  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
2 nfra1 2606 . . . . 5  |-  F/ x A. x  e.  d  pred ( x ,  A ,  R )  C_  d
32nfri 1754 . . . 4  |-  ( A. x  e.  d  pred ( x ,  A ,  R )  C_  d  ->  A. x A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
43bnj1352 29176 . . 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 ) )
54hbab 2287 . 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
) } )
61, 5hbxfreq 2399 1  |-  ( w  e.  B  ->  A. x  w  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556    C_ wss 3165    predc-bnj14 29029
This theorem is referenced by:  bnj1311  29370  bnj1373  29376  bnj1498  29407  bnj1525  29415  bnj1523  29417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561
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