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Theorem bnj1309 29052
Description: Technical lemma for bnj60 29092. 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 2593 . . . . 5  |-  F/ x A. x  e.  d  pred ( x ,  A ,  R )  C_  d
32nfri 1742 . . . 4  |-  ( A. x  e.  d  pred ( x ,  A ,  R )  C_  d  ->  A. x A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
43bnj1352 28860 . . 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 2274 . 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 2386 1  |-  ( w  e.  B  ->  A. x  w  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    C_ wss 3152    predc-bnj14 28713
This theorem is referenced by:  bnj1311  29054  bnj1373  29060  bnj1498  29091  bnj1525  29099  bnj1523  29101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548
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