MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psref Unicode version

Theorem psref 14317
Description: A poset is reflexive. (Contributed by NM, 13-May-2008.)
Hypothesis
Ref Expression
psref.1  |-  X  =  dom  R
Assertion
Ref Expression
psref  |-  ( ( R  e.  PosetRel  /\  A  e.  X )  ->  A R A )

Proof of Theorem psref
StepHypRef Expression
1 psref.1 . . . . 5  |-  X  =  dom  R
2 psdmrn 14316 . . . . . 6  |-  ( R  e.  PosetRel  ->  ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R ) )
32simpld 445 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  U. U. R )
41, 3syl5eq 2327 . . . 4  |-  ( R  e.  PosetRel  ->  X  =  U. U. R )
54eleq2d 2350 . . 3  |-  ( R  e.  PosetRel  ->  ( A  e.  X  <->  A  e.  U. U. R ) )
6 pslem 14315 . . . 4  |-  ( R  e.  PosetRel  ->  ( ( ( A R A  /\  A R A )  ->  A R A )  /\  ( A  e.  U. U. R  ->  A R A )  /\  ( ( A R A  /\  A R A )  ->  A  =  A )
) )
76simp2d 968 . . 3  |-  ( R  e.  PosetRel  ->  ( A  e. 
U. U. R  ->  A R A ) )
85, 7sylbid 206 . 2  |-  ( R  e.  PosetRel  ->  ( A  e.  X  ->  A R A ) )
98imp 418 1  |-  ( ( R  e.  PosetRel  /\  A  e.  X )  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   U.cuni 3827   class class class wbr 4023   dom cdm 4689   ran crn 4690   PosetRelcps 14301
This theorem is referenced by:  psss  14323  psssdm2  14324  spwpr4c  14341  ordtt1  17107
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ps 14306
  Copyright terms: Public domain W3C validator