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Mathbox for Norm Megill |
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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 5126 | . . 3 β’ ((π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π)) β π β π) | |
2 | 1 | 3ad2ant3 1136 | . 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 2996 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 class class class wbr 5106 βcfv 6497 (class class class)co 7358 lecple 17145 joincjn 18205 Atomscatm 37771 HLchlt 37858 LHypclh 38493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 |
This theorem is referenced by: cdlemeda 38807 |
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