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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemesner | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 13-Nov-2012.) |
Ref | Expression |
---|---|
cdlemesner.l | ⊢ ≤ = (le‘𝐾) |
cdlemesner.j | ⊢ ∨ = (join‘𝐾) |
cdlemesner.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemesner.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
cdlemesner | ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ≠ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbrne2 5162 | . . 3 ⊢ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑅 ≠ 𝑆) | |
2 | 1 | 3ad2ant3 1133 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≠ 𝑆) |
3 | 2 | necomd 2991 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ≠ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 lecple 17231 joincjn 18294 Atomscatm 38672 HLchlt 38759 LHypclh 39394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 |
This theorem is referenced by: cdlemeda 39708 |
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