Theorem pstr2 18750

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 18747	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))
2	1	ibi 270	. 2 ⊢ (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))
3	2	simp2d 1161	1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∩ cin 4089  ∪ cuni 4866   I cid 5544  ^◡ccnv 5649   ↾ cres 5652   ∘ ccom 5654  Rel wrel 5655  PosetRelcps 18743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-in 4091  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-res 5662  df-ps 18745

This theorem is used by:  pslem  18751  cnvps  18757  psss  18759  tsrdir  18783