MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tsrps Structured version   Visualization version   GIF version

Theorem tsrps 17202
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 2620 . . 3 dom 𝑅 = dom 𝑅
21istsr 17198 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
32simplbi 476 1 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988  cun 3565  wss 3567   × cxp 5102  ccnv 5103  dom cdm 5104  PosetRelcps 17179   TosetRel ctsr 17180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-dm 5114  df-tsr 17182
This theorem is referenced by:  cnvtsr  17203  tsrdir  17219  ordtbas2  20976  ordtrest2lem  20988  ordtrest2  20989  ordthauslem  21168  icopnfhmeo  22723  iccpnfhmeo  22725  xrhmeo  22726  cnvordtrestixx  29933  xrge0iifhmeo  29956
  Copyright terms: Public domain W3C validator