Theorem pstr2 18535

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 18532	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))
2	1	ibi 268	. 2 ⊢ (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))
3	2	simp2d 1149	1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ∩ cin 3889   ⊆ wss 3890  ∪ cuni 4845   I cid 5519  ^◡ccnv 5624   ↾ cres 5627   ∘ ccom 5629  Rel wrel 5630  PosetRelcps 18528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-in 3897  df-ss 3907  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-res 5637  df-ps 18530

This theorem is referenced by:  pslem  18536  cnvps  18542  psss  18544  tsrdir  18568