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Theorem pstr2 17145
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 17142 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
21ibi 256 . 2 (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅)))
32simp2d 1072 1 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  cin 3559  wss 3560   cuni 4409   I cid 4994  ccnv 5083  cres 5086  ccom 5088  Rel wrel 5089  PosetRelcps 17138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-v 3192  df-in 3567  df-ss 3574  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-res 5096  df-ps 17140
This theorem is referenced by:  pslem  17146  cnvps  17152  psss  17154  tsrdir  17178
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