Theorem psrel 18772

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 18771	. . 3 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))))
2	1	ibi 270	. 2 ⊢ (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴)))
3	2	simp1d 1160	1 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092  ∪ cuni 4870   I cid 5549  ^◡ccnv 5654   ↾ cres 5657   ∘ ccom 5659  Rel wrel 5660  PosetRelcps 18767

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-res 5667  df-ps 18769

This theorem is used by:  pslem  18775  cnvps  18781  psss  18783  cnvtsr  18791  tsrdir  18807