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| Mirrors > Home > MPE Home > Th. List > tospos | Structured version Visualization version GIF version | ||
| Description: A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
| Ref | Expression |
|---|---|
| tospos | ⊢ (𝐹 ∈ Toset → 𝐹 ∈ Poset) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
| 2 | eqid 2737 | . . 3 ⊢ (le‘𝐹) = (le‘𝐹) | |
| 3 | 1, 2 | istos 18463 | . 2 ⊢ (𝐹 ∈ Toset ↔ (𝐹 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑦 ∨ 𝑦(le‘𝐹)𝑥))) |
| 4 | 3 | simplbi 497 | 1 ⊢ (𝐹 ∈ Toset → 𝐹 ∈ Poset) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Posetcpo 18353 Tosetctos 18461 |
| 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 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-toset 18462 |
| This theorem is referenced by: tltnle 18467 resstos 32957 odutos 32958 tlt3 32960 xrsclat 33013 omndadd2d 33085 omndadd2rd 33086 omndmul2 33089 omndmul 33091 gsumle 33101 isarchi3 33194 archirngz 33196 archiabllem1a 33198 archiabllem2c 33202 orngsqr 33334 ofldchr 33344 ordtrest2NEWlem 33921 ordtrest2NEW 33922 ordtconnlem1 33923 toslat 48871 |
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