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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 5161 | . . 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 2990 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6537 (class class class)co 7405 lecple 17213 joincjn 18276 Atomscatm 38646 HLchlt 38733 LHypclh 39368 |
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 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 |
This theorem is referenced by: cdlemeda 39682 |
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