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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 17202 | . . 3 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴)))) | |
2 | 1 | ibi 256 | . 2 ⊢ (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))) |
3 | 2 | simp1d 1073 | 1 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 ∪ cuni 4436 I cid 5023 ◡ccnv 5113 ↾ cres 5116 ∘ ccom 5118 Rel wrel 5119 PosetRelcps 17198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-v 3202 df-in 3581 df-ss 3588 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-res 5126 df-ps 17200 |
This theorem is referenced by: pslem 17206 cnvps 17212 psss 17214 cnvtsr 17222 tsrdir 17238 |
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