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Theorem tsrps 17940
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 2738 . . 3 dom 𝑅 = dom 𝑅
21istsr 17936 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
32simplbi 501 1 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2113  cun 3839  wss 3841   × cxp 5517  ccnv 5518  dom cdm 5519  PosetRelcps 17917   TosetRel ctsr 17918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-rab 3062  df-v 3399  df-un 3846  df-in 3848  df-ss 3858  df-sn 4514  df-pr 4516  df-op 4520  df-br 5028  df-opab 5090  df-xp 5525  df-cnv 5527  df-dm 5529  df-tsr 17920
This theorem is referenced by:  cnvtsr  17941  tsrdir  17957  ordtbas2  21935  ordtrest2lem  21947  ordtrest2  21948  ordthauslem  22127  icopnfhmeo  23688  iccpnfhmeo  23690  xrhmeo  23691  cnvordtrestixx  31427  xrge0iifhmeo  31450
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