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Theorem pstr2 17810
Description: A poset is transitive. (Contributed by FL, 3-Aug-2009.)
Assertion
Ref Expression
pstr2 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)

Proof of Theorem pstr2
StepHypRef Expression
1 isps 17807 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
21ibi 270 . 2 (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅)))
32simp2d 1140 1 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2112  cin 3883  wss 3884   cuni 4803   I cid 5427  ccnv 5522  cres 5525  ccom 5527  Rel wrel 5528  PosetRelcps 17803
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-res 5535  df-ps 17805
This theorem is referenced by:  pslem  17811  cnvps  17817  psss  17819  tsrdir  17843
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