Theorem psrel 18579

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 18578	. . 3 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))))
2	1	ibi 267	. 2 ⊢ (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴)))
3	2	simp1d 1142	1 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ∩ cin 3925   ⊆ wss 3926  ∪ cuni 4883   I cid 5547  ^◡ccnv 5653   ↾ cres 5656   ∘ ccom 5658  Rel wrel 5659  PosetRelcps 18574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-in 3933  df-ss 3943  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-res 5666  df-ps 18576

This theorem is referenced by:  pslem  18582  cnvps  18588  psss  18590  cnvtsr  18598  tsrdir  18614