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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 36422 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 ))) |
6 | orcom 866 | . . . . . 6 ⊢ ((𝑋 = 𝑃 ∨ 𝑋 = 0 ) ↔ (𝑋 = 0 ∨ 𝑋 = 𝑃)) | |
7 | neor 3108 | . . . . . 6 ⊢ ((𝑋 = 0 ∨ 𝑋 = 𝑃) ↔ (𝑋 ≠ 0 → 𝑋 = 𝑃)) | |
8 | 6, 7 | bitri 277 | . . . . 5 ⊢ ((𝑋 = 𝑃 ∨ 𝑋 = 0 ) ↔ (𝑋 ≠ 0 → 𝑋 = 𝑃)) |
9 | 5, 8 | syl6bb 289 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 ≠ 0 → 𝑋 = 𝑃))) |
10 | 9 | biimpd 231 | . . 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 421 | 1 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 0 ∧ 𝑋 ≤ 𝑃)) → 𝑋 = 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5059 ‘cfv 6350 Basecbs 16477 lecple 16566 0.cp0 17641 OPcops 36302 Atomscatm 36393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-proset 17532 df-poset 17550 df-plt 17562 df-glb 17579 df-p0 17643 df-oposet 36306 df-covers 36396 df-ats 36397 |
This theorem is referenced by: dalemcea 36790 cdlemg12g 37779 |
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