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| Mirrors > Home > MPE Home > Th. List > psrel | Structured version Visualization version GIF version | ||
| Description: A poset is a relation. (Contributed by NM, 12-May-2008.) |
| Ref | Expression |
|---|---|
| psrel | ⊢ (𝐴 ∈ PosetRel → Rel 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isps 18613 | . . 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 1143 | 1 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ∪ cuni 4907 I cid 5577 ◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 Rel wrel 5690 PosetRelcps 18609 |
| 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 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-res 5697 df-ps 18611 |
| This theorem is referenced by: pslem 18617 cnvps 18623 psss 18625 cnvtsr 18633 tsrdir 18649 |
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