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Mirrors > Home > MPE Home > Th. List > postr | Structured version Visualization version GIF version |
Description: A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.) |
Ref | Expression |
---|---|
posi.b | β’ π΅ = (BaseβπΎ) |
posi.l | β’ β€ = (leβπΎ) |
Ref | Expression |
---|---|
postr | β’ ((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β€ π β§ π β€ π) β π β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | posi.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | posi.l | . . 3 β’ β€ = (leβπΎ) | |
3 | 1, 2 | posi 18270 | . 2 β’ ((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β€ π β§ ((π β€ π β§ π β€ π) β π = π) β§ ((π β€ π β§ π β€ π) β π β€ π))) |
4 | 3 | simp3d 1145 | 1 β’ ((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β€ π β§ π β€ π) β π β€ π)) |
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 5149 βcfv 6544 Basecbs 17144 lecple 17204 Posetcpo 18260 |
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-rex 3072 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-iota 6496 df-fv 6552 df-poset 18266 |
This theorem is referenced by: odupos 18281 plttr 18295 joinle 18339 meetle 18353 lattr 18397 omndadd2d 32226 omndadd2rd 32227 omndmul2 32230 atlatle 38190 cvratlem 38292 llncmp 38393 llncvrlpln 38429 lplncmp 38433 lplncvrlvol 38487 lvolcmp 38488 pmaple 38632 |
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