Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1418 Structured version   Unicode version

Theorem bnj1418 29346
Description: Property of  pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1418  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )

Proof of Theorem bnj1418
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 breq1 4207 . 2  |-  ( z  =  y  ->  (
z R x  <->  y R x ) )
2 df-bnj14 28990 . . 3  |-  pred (
x ,  A ,  R )  =  {
z  e.  A  | 
z R x }
32bnj1538 29163 . 2  |-  ( z  e.  pred ( x ,  A ,  R )  ->  z R x )
41, 3vtoclga 3009 1  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   class class class wbr 4204    predc-bnj14 28989
This theorem is referenced by:  bnj1417  29347  bnj1523  29377
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