Theorem pstr2 18494

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 18491	. . 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 2113   ∩ cin 3900   ⊆ wss 3901  ∪ cuni 4863   I cid 5518  ^◡ccnv 5623   ↾ cres 5626   ∘ ccom 5628  Rel wrel 5629  PosetRelcps 18487

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 2115  ax-9 2123  ax-ext 2708

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 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-in 3908  df-ss 3918  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-res 5636  df-ps 18489

This theorem is referenced by:  pslem  18495  cnvps  18501  psss  18503  tsrdir  18527