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Theorem bnj1309 29109
Description: Technical lemma for bnj60 29149. 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 hbra1 2723 . . . 4  |-  ( A. x  e.  d  pred ( x ,  A ,  R )  C_  d  ->  A. x A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
32bnj1352 28917 . . 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 ) )
43hbab 2403 . 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
) } )
51, 4hbxfreq 2515 1  |-  ( w  e.  B  ->  A. x  w  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   {cab 2398   A.wral 2674    C_ wss 3288    predc-bnj14 28770
This theorem is referenced by:  bnj1311  29111  bnj1373  29117  bnj1498  29148  bnj1525  29156  bnj1523  29158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-ral 2679
  Copyright terms: Public domain W3C validator