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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > oplecon1b | Structured version Visualization version GIF version |
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 30754 analog.) (Contributed by NM, 6-Nov-2011.) |
Ref | Expression |
---|---|
opcon3.b | β’ π΅ = (BaseβπΎ) |
opcon3.l | β’ β€ = (leβπΎ) |
opcon3.o | β’ β₯ = (ocβπΎ) |
Ref | Expression |
---|---|
oplecon1b | β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β (( β₯ βπ) β€ π β ( β₯ βπ) β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opcon3.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | opcon3.o | . . . . 5 β’ β₯ = (ocβπΎ) | |
3 | 1, 2 | opoccl 38064 | . . . 4 β’ ((πΎ β OP β§ π β π΅) β ( β₯ βπ) β π΅) |
4 | 3 | 3adant3 1133 | . . 3 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β ( β₯ βπ) β π΅) |
5 | opcon3.l | . . . 4 β’ β€ = (leβπΎ) | |
6 | 1, 5, 2 | oplecon3b 38070 | . . 3 β’ ((πΎ β OP β§ ( β₯ βπ) β π΅ β§ π β π΅) β (( β₯ βπ) β€ π β ( β₯ βπ) β€ ( β₯ β( β₯ βπ)))) |
7 | 4, 6 | syld3an2 1412 | . 2 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β (( β₯ βπ) β€ π β ( β₯ βπ) β€ ( β₯ β( β₯ βπ)))) |
8 | 1, 2 | opococ 38065 | . . . 4 β’ ((πΎ β OP β§ π β π΅) β ( β₯ β( β₯ βπ)) = π) |
9 | 8 | 3adant3 1133 | . . 3 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β ( β₯ β( β₯ βπ)) = π) |
10 | 9 | breq2d 5161 | . 2 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β (( β₯ βπ) β€ ( β₯ β( β₯ βπ)) β ( β₯ βπ) β€ π)) |
11 | 7, 10 | bitrd 279 | 1 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β (( β₯ βπ) β€ π β ( β₯ βπ) β€ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5149 βcfv 6544 Basecbs 17144 lecple 17204 occoc 17205 OPcops 38042 |
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 ax-nul 5307 |
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 2942 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-dm 5687 df-iota 6496 df-fv 6552 df-ov 7412 df-oposet 38046 |
This theorem is referenced by: opoc1 38072 oldmm1 38087 |
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