Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1152 Structured version   Unicode version

Theorem bnj1152 29304
Description: Technical lemma for bnj69 29316. 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.)
Assertion
Ref Expression
bnj1152  |-  ( Y  e.  pred ( X ,  A ,  R )  <->  ( Y  e.  A  /\  Y R X ) )

Proof of Theorem bnj1152
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 breq1 4207 . 2  |-  ( y  =  Y  ->  (
y R X  <->  Y R X ) )
2 df-bnj14 28990 . 2  |-  pred ( X ,  A ,  R )  =  {
y  e.  A  | 
y R X }
31, 2elrab2 3086 1  |-  ( Y  e.  pred ( X ,  A ,  R )  <->  ( Y  e.  A  /\  Y R X ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725   class class class wbr 4204    predc-bnj14 28989
This theorem is referenced by:  bnj1175  29310  bnj1177  29312  bnj1388  29339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-bnj14 28990
  Copyright terms: Public domain W3C validator