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Theorem tsrps 14423
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 2358 . . 3  |-  dom  R  =  dom  R
21istsr 14419 . 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 1710    u. cun 3226    C_ wss 3228    X. cxp 4766   `'ccnv 4767   dom cdm 4768   PosetRelcps 14394    TosetRel ctsr 14395
This theorem is referenced by:  cnvtsr  14424  tsrdir  14453  ordtbas2  17021  ordtrest2lem  17033  ordtrest2  17034  ordthauslem  17211  icopnfhmeo  18539  iccpnfhmeo  18541  xrhmeo  18542  cnvordtrestixx  23467  xrge0iifhmeo  23478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-xp 4774  df-cnv 4776  df-dm 4778  df-tsr 14400
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