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Theorem tsrps 14616
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 2412 . . 3  |-  dom  R  =  dom  R
21istsr 14612 . 2  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
32simplbi 447 1  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721    u. cun 3286    C_ wss 3288    X. cxp 4843   `'ccnv 4844   dom cdm 4845   PosetRelcps 14587    TosetRel ctsr 14588
This theorem is referenced by:  cnvtsr  14617  tsrdir  14646  ordtbas2  17217  ordtrest2lem  17229  ordtrest2  17230  ordthauslem  17409  icopnfhmeo  18929  iccpnfhmeo  18931  xrhmeo  18932  cnvordtrestixx  24272  xrge0iifhmeo  24283
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
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-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-cnv 4853  df-dm 4855  df-tsr 14593
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