Theorem pstr2 18474

Description: A poset is transitive. (Contributed by FL, 3-Aug-2009.)

Assertion

Ref	Expression
pstr2	⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)

Proof of Theorem pstr2

Step	Hyp	Ref	Expression
1		isps 18471	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))
2	1	ibi 267	. 2 ⊢ (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))
3	2	simp2d 1143	1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ∩ cin 3901   ⊆ wss 3902  ∪ cuni 4859   I cid 5510  ^◡ccnv 5615   ↾ cres 5618   ∘ ccom 5620  Rel wrel 5621  PosetRelcps 18467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-in 3909  df-ss 3919  df-uni 4860  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-res 5628  df-ps 18469

This theorem is referenced by:  pslem  18475  cnvps  18481  psss  18483  tsrdir  18507