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Theorem tsrps 14658
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 2438 . . 3  |-  dom  R  =  dom  R
21istsr 14654 . 2  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
32simplbi 448 1  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726    u. cun 3320    C_ wss 3322    X. cxp 4879   `'ccnv 4880   dom cdm 4881   PosetRelcps 14629    TosetRel ctsr 14630
This theorem is referenced by:  cnvtsr  14659  tsrdir  14688  ordtbas2  17260  ordtrest2lem  17272  ordtrest2  17273  ordthauslem  17452  icopnfhmeo  18973  iccpnfhmeo  18975  xrhmeo  18976  cnvordtrestixx  24316  xrge0iifhmeo  24327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-cnv 4889  df-dm 4891  df-tsr 14635
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