Theorem psrel 17813

Description: A poset is a relation. (Contributed by NM, 12-May-2008.)

Assertion

Ref	Expression
psrel	⊢ (𝐴 ∈ PosetRel → Rel 𝐴)

Proof of Theorem psrel

Step	Hyp	Ref	Expression
1		isps 17812	. . 3 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))))
2	1	ibi 269	. 2 ⊢ (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴)))
3	2	simp1d 1138	1 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ∩ cin 3935   ⊆ wss 3936  ∪ cuni 4838   I cid 5459  ^◡ccnv 5554   ↾ cres 5557   ∘ ccom 5559  Rel wrel 5560  PosetRelcps 17808

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-in 3943  df-ss 3952  df-uni 4839  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-res 5567  df-ps 17810

This theorem is referenced by:  pslem  17816  cnvps  17822  psss  17824  cnvtsr  17832  tsrdir  17848