Theorem psrel 18384

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 18383	. . 3 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))))
2	1	ibi 266	. 2 ⊢ (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴)))
3	2	simp1d 1141	1 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ∩ cin 3897   ⊆ wss 3898  ∪ cuni 4852   I cid 5517  ^◡ccnv 5619   ↾ cres 5622   ∘ ccom 5624  Rel wrel 5625  PosetRelcps 18379

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-in 3905  df-ss 3915  df-uni 4853  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-res 5632  df-ps 18381

This theorem is referenced by:  pslem  18387  cnvps  18393  psss  18395  cnvtsr  18403  tsrdir  18419