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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 17804 | . . 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 1139 | 1 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 ∪ cuni 4800 I cid 5424 ◡ccnv 5518 ↾ cres 5521 ∘ ccom 5523 Rel wrel 5524 PosetRelcps 17800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-res 5531 df-ps 17802 |
This theorem is referenced by: pslem 17808 cnvps 17814 psss 17816 cnvtsr 17824 tsrdir 17840 |
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