Theorem psrel 18535

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 18534	. . 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 2109   ∩ cin 3916   ⊆ wss 3917  ∪ cuni 4874   I cid 5535  ^◡ccnv 5640   ↾ cres 5643   ∘ ccom 5645  Rel wrel 5646  PosetRelcps 18530

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-in 3924  df-ss 3934  df-uni 4875  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-res 5653  df-ps 18532

This theorem is referenced by:  pslem  18538  cnvps  18544  psss  18546  cnvtsr  18554  tsrdir  18570