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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 17592 | . . 3 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴)))) | |
2 | 1 | ibi 259 | . 2 ⊢ (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))) |
3 | 2 | simp1d 1133 | 1 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 ⊆ wss 3792 ∪ cuni 4673 I cid 5262 ◡ccnv 5356 ↾ cres 5359 ∘ ccom 5361 Rel wrel 5362 PosetRelcps 17588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rex 3096 df-v 3400 df-in 3799 df-ss 3806 df-uni 4674 df-br 4889 df-opab 4951 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-res 5369 df-ps 17590 |
This theorem is referenced by: pslem 17596 cnvps 17602 psss 17604 cnvtsr 17612 tsrdir 17628 |
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