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Theorem bnj1309 33811
Description: Technical lemma for bnj60 33851. 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 2825 . . . 4  |-  ( A. x  e.  d  pred ( x ,  A ,  R )  C_  d  ->  A. x A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
32bnj1352 33619 . . 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 2433 . 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 2565 1  |-  ( w  e.  B  ->  A. x  w  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1381    = wceq 1383    e. wcel 1804   {cab 2428   A.wral 2793    C_ wss 3461    predc-bnj14 33473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-ral 2798
This theorem is referenced by:  bnj1311  33813  bnj1373  33819  bnj1498  33850  bnj1525  33858  bnj1523  33860
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