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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tlt2 | Structured version Visualization version GIF version |
Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
Ref | Expression |
---|---|
tlt2.b | β’ π΅ = (BaseβπΎ) |
tlt2.e | β’ β€ = (leβπΎ) |
tlt2.l | β’ < = (ltβπΎ) |
Ref | Expression |
---|---|
tlt2 | β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π β€ π β¨ π < π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidd 894 | . 2 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π β€ π β¨ Β¬ π β€ π)) | |
2 | tlt2.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
3 | tlt2.e | . . . . 5 β’ β€ = (leβπΎ) | |
4 | tlt2.l | . . . . 5 β’ < = (ltβπΎ) | |
5 | 2, 3, 4 | tltnle 18371 | . . . 4 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π < π β Β¬ π β€ π)) |
6 | 5 | 3com23 1126 | . . 3 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π < π β Β¬ π β€ π)) |
7 | 6 | orbi2d 914 | . 2 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β ((π β€ π β¨ π < π) β (π β€ π β¨ Β¬ π β€ π))) |
8 | 1, 7 | mpbird 256 | 1 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π β€ π β¨ π < π)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β¨ wo 845 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5147 βcfv 6540 Basecbs 17140 lecple 17200 ltcplt 18257 Tosetctos 18365 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-proset 18244 df-poset 18262 df-plt 18279 df-toset 18366 |
This theorem is referenced by: tlt3 32127 archirngz 32322 archiabllem2a 32327 |
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