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

Theorem tsrps 17581
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 2825 . . 3 dom 𝑅 = dom 𝑅
21istsr 17577 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
32simplbi 493 1 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2164  cun 3796  wss 3798   × cxp 5344  ccnv 5345  dom cdm 5346  PosetRelcps 17558   TosetRel ctsr 17559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-cnv 5354  df-dm 5356  df-tsr 17561
This theorem is referenced by:  cnvtsr  17582  tsrdir  17598  ordtbas2  21373  ordtrest2lem  21385  ordtrest2  21386  ordthauslem  21565  icopnfhmeo  23119  iccpnfhmeo  23121  xrhmeo  23122  cnvordtrestixx  30500  xrge0iifhmeo  30523
  Copyright terms: Public domain W3C validator