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Theorem tsrps 14330
Description: A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
Assertion
Ref Expression
tsrps  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )

Proof of Theorem tsrps
StepHypRef Expression
1 eqid 2283 . . 3  |-  dom  R  =  dom  R
21istsr 14326 . 2  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
32simplbi 446 1  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    u. cun 3150    C_ wss 3152    X. cxp 4687   `'ccnv 4688   dom cdm 4689   PosetRelcps 14301    TosetRel ctsr 14302
This theorem is referenced by:  cnvtsr  14331  tsrdir  14360  ordtbas2  16921  ordtrest2lem  16933  ordtrest2  16934  ordthauslem  17111  icopnfhmeo  18441  iccpnfhmeo  18443  xrhmeo  18444  cnvordtrestixx  23297  xrge0iifhmeo  23318  supnuf  25629  supexr  25631
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  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-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-tsr 14307
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