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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 895 | . 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 18316 | . . . 4 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π < π β Β¬ π β€ π)) |
6 | 5 | 3com23 1127 | . . 3 β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π < π β Β¬ π β€ π)) |
7 | 6 | orbi2d 915 | . 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 257 | 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 846 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5106 βcfv 6497 Basecbs 17088 lecple 17145 ltcplt 18202 Tosetctos 18310 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-proset 18189 df-poset 18207 df-plt 18224 df-toset 18311 |
This theorem is referenced by: tlt3 31879 archirngz 32074 archiabllem2a 32079 |
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