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Theorem pstr2 18459
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 18456 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
21ibi 266 . 2 (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅)))
32simp2d 1143 1 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cin 3909  wss 3910   cuni 4865   I cid 5530  ccnv 5632  cres 5635  ccom 5637  Rel wrel 5638  PosetRelcps 18452
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-in 3917  df-ss 3927  df-uni 4866  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-res 5645  df-ps 18454
This theorem is referenced by:  pslem  18460  cnvps  18466  psss  18468  tsrdir  18492
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