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Theorem tsrps 18570
Description: A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
Assertion
Ref Expression
tsrps (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)

Proof of Theorem tsrps
StepHypRef Expression
1 eqid 2727 . . 3 dom 𝑅 = dom 𝑅
21istsr 18566 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
32simplbi 497 1 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  cun 3942  wss 3944   × cxp 5670  ccnv 5671  dom cdm 5672  PosetRelcps 18547   TosetRel ctsr 18548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-tsr 18550
This theorem is referenced by:  cnvtsr  18571  tsrdir  18587  ordtbas2  23082  ordtrest2lem  23094  ordtrest2  23095  ordthauslem  23274  icopnfhmeo  24855  iccpnfhmeo  24857  xrhmeo  24858  cnvordtrestixx  33450  xrge0iifhmeo  33473
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