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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > leat2 | Structured version Visualization version GIF version |
Description: A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.) |
Ref | Expression |
---|---|
leatom.b | β’ π΅ = (BaseβπΎ) |
leatom.l | β’ β€ = (leβπΎ) |
leatom.z | β’ 0 = (0.βπΎ) |
leatom.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
leat2 | β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ (π β 0 β§ π β€ π)) β π = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leatom.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
2 | leatom.l | . . . . . 6 β’ β€ = (leβπΎ) | |
3 | leatom.z | . . . . . 6 β’ 0 = (0.βπΎ) | |
4 | leatom.a | . . . . . 6 β’ π΄ = (AtomsβπΎ) | |
5 | 1, 2, 3, 4 | leatb 37800 | . . . . 5 β’ ((πΎ β OP β§ π β π΅ β§ π β π΄) β (π β€ π β (π = π β¨ π = 0 ))) |
6 | orcom 869 | . . . . . 6 β’ ((π = π β¨ π = 0 ) β (π = 0 β¨ π = π)) | |
7 | neor 3033 | . . . . . 6 β’ ((π = 0 β¨ π = π) β (π β 0 β π = π)) | |
8 | 6, 7 | bitri 275 | . . . . 5 β’ ((π = π β¨ π = 0 ) β (π β 0 β π = π)) |
9 | 5, 8 | bitrdi 287 | . . . 4 β’ ((πΎ β OP β§ π β π΅ β§ π β π΄) β (π β€ π β (π β 0 β π = π))) |
10 | 9 | biimpd 228 | . . 3 β’ ((πΎ β OP β§ π β π΅ β§ π β π΄) β (π β€ π β (π β 0 β π = π))) |
11 | 10 | com23 86 | . 2 β’ ((πΎ β OP β§ π β π΅ β§ π β π΄) β (π β 0 β (π β€ π β π = π))) |
12 | 11 | imp32 420 | 1 β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ (π β 0 β§ π β€ π)) β π = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β¨ wo 846 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 class class class wbr 5106 βcfv 6497 Basecbs 17088 lecple 17145 0.cp0 18317 OPcops 37680 Atomscatm 37771 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 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-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-proset 18189 df-poset 18207 df-plt 18224 df-glb 18241 df-p0 18319 df-oposet 37684 df-covers 37774 df-ats 37775 |
This theorem is referenced by: dalemcea 38169 cdlemg12g 39158 |
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