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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opnoncon | Structured version Visualization version GIF version |
Description: Law of contradiction for orthoposets. (chocin 30735 analog.) (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
opnoncon.b | β’ π΅ = (BaseβπΎ) |
opnoncon.o | β’ β₯ = (ocβπΎ) |
opnoncon.m | β’ β§ = (meetβπΎ) |
opnoncon.z | β’ 0 = (0.βπΎ) |
Ref | Expression |
---|---|
opnoncon | β’ ((πΎ β OP β§ π β π΅) β (π β§ ( β₯ βπ)) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opnoncon.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2732 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
3 | opnoncon.o | . . . 4 β’ β₯ = (ocβπΎ) | |
4 | eqid 2732 | . . . 4 β’ (joinβπΎ) = (joinβπΎ) | |
5 | opnoncon.m | . . . 4 β’ β§ = (meetβπΎ) | |
6 | opnoncon.z | . . . 4 β’ 0 = (0.βπΎ) | |
7 | eqid 2732 | . . . 4 β’ (1.βπΎ) = (1.βπΎ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 38040 | . . 3 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β ((( β₯ βπ) β π΅ β§ ( β₯ β( β₯ βπ)) = π β§ (π(leβπΎ)π β ( β₯ βπ)(leβπΎ)( β₯ βπ))) β§ (π(joinβπΎ)( β₯ βπ)) = (1.βπΎ) β§ (π β§ ( β₯ βπ)) = 0 )) |
9 | 8 | 3anidm23 1421 | . 2 β’ ((πΎ β OP β§ π β π΅) β ((( β₯ βπ) β π΅ β§ ( β₯ β( β₯ βπ)) = π β§ (π(leβπΎ)π β ( β₯ βπ)(leβπΎ)( β₯ βπ))) β§ (π(joinβπΎ)( β₯ βπ)) = (1.βπΎ) β§ (π β§ ( β₯ βπ)) = 0 )) |
10 | 9 | simp3d 1144 | 1 β’ ((πΎ β OP β§ π β π΅) β (π β§ ( β₯ βπ)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5147 βcfv 6540 (class class class)co 7405 Basecbs 17140 lecple 17200 occoc 17201 joincjn 18260 meetcmee 18261 0.cp0 18372 1.cp1 18373 OPcops 38030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-dm 5685 df-iota 6492 df-fv 6548 df-ov 7408 df-oposet 38034 |
This theorem is referenced by: omlfh1N 38116 omlspjN 38119 atlatmstc 38177 pnonsingN 38792 lhpocnle 38875 dochnoncon 40250 |
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