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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 | β’ β€ = (leβπΎ) |
cdlemesner.j | β’ β¨ = (joinβπΎ) |
cdlemesner.a | β’ π΄ = (AtomsβπΎ) |
cdlemesner.h | β’ π» = (LHypβπΎ) |
Ref | Expression |
---|---|
cdlemesner | β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ (π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π))) β π β π ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbrne2 5163 | . . 3 β’ ((π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π)) β π β π) | |
2 | 1 | 3ad2ant3 1132 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ (π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π))) β π β π) |
3 | 2 | necomd 2986 | 1 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ (π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π))) β π β π ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 class class class wbr 5143 βcfv 6543 (class class class)co 7416 lecple 17239 joincjn 18302 Atomscatm 38791 HLchlt 38878 LHypclh 39513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 |
This theorem is referenced by: cdlemeda 39827 |
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