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Theorem psref 14333
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 14332 . . . . . 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 2340 . . . 4  |-  ( R  e.  PosetRel  ->  X  =  U. U. R )
54eleq2d 2363 . . 3  |-  ( R  e.  PosetRel  ->  ( A  e.  X  <->  A  e.  U. U. R ) )
6 pslem 14331 . . . 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 1632    e. wcel 1696   U.cuni 3843   class class class wbr 4039   dom cdm 4705   ran crn 4706   PosetRelcps 14317
This theorem is referenced by:  psss  14339  psssdm2  14340  spwpr4c  14357  ordtt1  17123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ps 14322
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