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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tlt3 | Structured version Visualization version GIF version |
Description: In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
Ref | Expression |
---|---|
tlt3.b | β’ π΅ = (BaseβπΎ) |
tlt3.l | β’ < = (ltβπΎ) |
Ref | Expression |
---|---|
tlt3 | β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π = π β¨ π < π β¨ π < π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tlt3.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2728 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
3 | tlt3.l | . . . 4 β’ < = (ltβπΎ) | |
4 | 1, 2, 3 | tlt2 32690 | . . 3 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π(leβπΎ)π β¨ π < π)) |
5 | tospos 18405 | . . . . 5 β’ (πΎ β Toset β πΎ β Poset) | |
6 | 1, 2, 3 | pleval2 18322 | . . . . . 6 β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β (π(leβπΎ)π β (π < π β¨ π = π))) |
7 | orcom 869 | . . . . . 6 β’ ((π < π β¨ π = π) β (π = π β¨ π < π)) | |
8 | 6, 7 | bitrdi 287 | . . . . 5 β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β (π(leβπΎ)π β (π = π β¨ π < π))) |
9 | 5, 8 | syl3an1 1161 | . . . 4 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π(leβπΎ)π β (π = π β¨ π < π))) |
10 | 9 | orbi1d 915 | . . 3 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β ((π(leβπΎ)π β¨ π < π) β ((π = π β¨ π < π) β¨ π < π))) |
11 | 4, 10 | mpbid 231 | . 2 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β ((π = π β¨ π < π) β¨ π < π)) |
12 | df-3or 1086 | . 2 β’ ((π = π β¨ π < π β¨ π < π) β ((π = π β¨ π < π) β¨ π < π)) | |
13 | 11, 12 | sylibr 233 | 1 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π = π β¨ π < π β¨ π < π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β¨ wo 846 β¨ w3o 1084 β§ w3a 1085 = wceq 1534 β wcel 2099 class class class wbr 5142 βcfv 6542 Basecbs 17173 lecple 17233 Posetcpo 18292 ltcplt 18293 Tosetctos 18401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-proset 18280 df-poset 18298 df-plt 18315 df-toset 18402 |
This theorem is referenced by: archirngz 32891 archiabllem1b 32894 archiabllem2b 32898 |
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