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Theorem tsrps 18539
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 2732 . . 3 dom 𝑅 = dom 𝑅
21istsr 18535 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
32simplbi 498 1 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cun 3946  wss 3948   × cxp 5674  ccnv 5675  dom cdm 5676  PosetRelcps 18516   TosetRel ctsr 18517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-tsr 18519
This theorem is referenced by:  cnvtsr  18540  tsrdir  18556  ordtbas2  22694  ordtrest2lem  22706  ordtrest2  22707  ordthauslem  22886  icopnfhmeo  24458  iccpnfhmeo  24460  xrhmeo  24461  cnvordtrestixx  32888  xrge0iifhmeo  32911
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