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Theorem tsrps 15699
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 2462 . . 3  |-  dom  R  =  dom  R
21istsr 15695 . 2  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
32simplbi 460 1  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1762    u. cun 3469    C_ wss 3471    X. cxp 4992   `'ccnv 4993   dom cdm 4994   PosetRelcps 15676    TosetRel ctsr 15677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-cnv 5002  df-dm 5004  df-tsr 15679
This theorem is referenced by:  cnvtsr  15700  tsrdir  15716  ordtbas2  19453  ordtrest2lem  19465  ordtrest2  19466  ordthauslem  19645  icopnfhmeo  21173  iccpnfhmeo  21175  xrhmeo  21176  cnvordtrestixx  27519  xrge0iifhmeo  27542
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