| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > leat2 | Structured version Visualization version GIF version | ||
| Description: A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.) |
| Ref | Expression |
|---|---|
| leatom.b | ⊢ 𝐵 = (Base‘𝐾) |
| leatom.l | ⊢ ≤ = (le‘𝐾) |
| leatom.z | ⊢ 0 = (0.‘𝐾) |
| leatom.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| Ref | Expression |
|---|---|
| leat2 | ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 0 ∧ 𝑋 ≤ 𝑃)) → 𝑋 = 𝑃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | leatom.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | leatom.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
| 3 | leatom.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
| 4 | leatom.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 5 | 1, 2, 3, 4 | leatb 39293 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 ))) |
| 6 | orcom 871 | . . . . . 6 ⊢ ((𝑋 = 𝑃 ∨ 𝑋 = 0 ) ↔ (𝑋 = 0 ∨ 𝑋 = 𝑃)) | |
| 7 | neor 3034 | . . . . . 6 ⊢ ((𝑋 = 0 ∨ 𝑋 = 𝑃) ↔ (𝑋 ≠ 0 → 𝑋 = 𝑃)) | |
| 8 | 6, 7 | bitri 275 | . . . . 5 ⊢ ((𝑋 = 𝑃 ∨ 𝑋 = 0 ) ↔ (𝑋 ≠ 0 → 𝑋 = 𝑃)) |
| 9 | 5, 8 | bitrdi 287 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 ≠ 0 → 𝑋 = 𝑃))) |
| 10 | 9 | biimpd 229 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 → (𝑋 ≠ 0 → 𝑋 = 𝑃))) |
| 11 | 10 | com23 86 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≠ 0 → (𝑋 ≤ 𝑃 → 𝑋 = 𝑃))) |
| 12 | 11 | imp32 418 | 1 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 0 ∧ 𝑋 ≤ 𝑃)) → 𝑋 = 𝑃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 0.cp0 18468 OPcops 39173 Atomscatm 39264 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-proset 18340 df-poset 18359 df-plt 18375 df-glb 18392 df-p0 18470 df-oposet 39177 df-covers 39267 df-ats 39268 |
| This theorem is referenced by: dalemcea 39662 cdlemg12g 40651 |
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