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Theorem psrel 18568

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 18567	. . 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 1139	1 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∩ cin 3948 ⊆ wss 3949 ∪ cuni 4912 I cid 5579 ^◡ccnv 5681 ↾ cres 5684 ∘ ccom 5686 Rel wrel 5687 PosetRelcps 18563

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699

This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-in 3956 df-ss 3966 df-uni 4913 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-res 5694 df-ps 18565

This theorem is referenced by: pslem 18571 cnvps 18577 psss 18579 cnvtsr 18587 tsrdir 18603

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