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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opexmid | Structured version Visualization version GIF version |
Description: Law of excluded middle for orthoposets. (chjo 30460 analog.) (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
opexmid.b | β’ π΅ = (BaseβπΎ) |
opexmid.o | β’ β₯ = (ocβπΎ) |
opexmid.j | β’ β¨ = (joinβπΎ) |
opexmid.u | β’ 1 = (1.βπΎ) |
Ref | Expression |
---|---|
opexmid | β’ ((πΎ β OP β§ π β π΅) β (π β¨ ( β₯ βπ)) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opexmid.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2737 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
3 | opexmid.o | . . . 4 β’ β₯ = (ocβπΎ) | |
4 | opexmid.j | . . . 4 β’ β¨ = (joinβπΎ) | |
5 | eqid 2737 | . . . 4 β’ (meetβπΎ) = (meetβπΎ) | |
6 | eqid 2737 | . . . 4 β’ (0.βπΎ) = (0.βπΎ) | |
7 | opexmid.u | . . . 4 β’ 1 = (1.βπΎ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 37647 | . . 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 β¨ ( β₯ βπ)) = 1 β§ (π(meetβπΎ)( β₯ βπ)) = (0.βπΎ))) |
9 | 8 | 3anidm23 1422 | . 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 β¨ ( β₯ βπ)) = 1 β§ (π(meetβπΎ)( β₯ βπ)) = (0.βπΎ))) |
10 | 9 | simp2d 1144 | 1 β’ ((πΎ β OP β§ π β π΅) β (π β¨ ( β₯ βπ)) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17084 lecple 17141 occoc 17142 joincjn 18201 meetcmee 18202 0.cp0 18313 1.cp1 18314 OPcops 37637 |
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 2708 ax-nul 5264 |
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 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rab 3409 df-v 3448 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-uni 4867 df-br 5107 df-dm 5644 df-iota 6449 df-fv 6505 df-ov 7361 df-oposet 37641 |
This theorem is referenced by: dih1 39752 |
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