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Theorem psref 14568
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 14567 . . . . . 6  |-  ( R  e.  PosetRel  ->  ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R ) )
32simpld 446 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  U. U. R )
41, 3syl5eq 2432 . . . 4  |-  ( R  e.  PosetRel  ->  X  =  U. U. R )
54eleq2d 2455 . . 3  |-  ( R  e.  PosetRel  ->  ( A  e.  X  <->  A  e.  U. U. R ) )
6 pslem 14566 . . . 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 970 . . 3  |-  ( R  e.  PosetRel  ->  ( A  e. 
U. U. R  ->  A R A ) )
85, 7sylbid 207 . 2  |-  ( R  e.  PosetRel  ->  ( A  e.  X  ->  A R A ) )
98imp 419 1  |-  ( ( R  e.  PosetRel  /\  A  e.  X )  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   U.cuni 3958   class class class wbr 4154   dom cdm 4819   ran crn 4820   PosetRelcps 14552
This theorem is referenced by:  psss  14574  psssdm2  14575  spwpr4c  14592  ordtt1  17366
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ps 14557
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