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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 30792 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 38150 | . . . 4 β’ ((πΎ β OP β§ π β π΅) β ( β₯ βπ) β π΅) |
4 | 3 | 3adant3 1132 | . . 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 38156 | . . 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 1411 | . 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 38151 | . . . 4 β’ ((πΎ β OP β§ π β π΅) β ( β₯ β( β₯ βπ)) = π) |
9 | 8 | 3adant3 1132 | . . 3 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β ( β₯ β( β₯ βπ)) = π) |
10 | 9 | breq2d 5160 | . 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 278 | 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 1087 = wceq 1541 β wcel 2106 class class class wbr 5148 βcfv 6543 Basecbs 17146 lecple 17206 occoc 17207 OPcops 38128 |
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 5306 |
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 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-dm 5686 df-iota 6495 df-fv 6551 df-ov 7414 df-oposet 38132 |
This theorem is referenced by: opoc1 38158 oldmm1 38173 |
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