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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > leat | Structured version Visualization version GIF version |
Description: A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-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 |
---|---|
leat | β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ π β€ π) β (π = π β¨ π = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leatom.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | leatom.l | . . 3 β’ β€ = (leβπΎ) | |
3 | leatom.z | . . 3 β’ 0 = (0.βπΎ) | |
4 | leatom.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
5 | 1, 2, 3, 4 | leatb 38157 | . 2 β’ ((πΎ β OP β§ π β π΅ β§ π β π΄) β (π β€ π β (π = π β¨ π = 0 ))) |
6 | 5 | biimpa 477 | 1 β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ π β€ π) β (π = π β¨ π = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β¨ wo 845 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5148 βcfv 6543 Basecbs 17143 lecple 17203 0.cp0 18375 OPcops 38037 Atomscatm 38128 |
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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
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-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-proset 18247 df-poset 18265 df-plt 18282 df-glb 18299 df-p0 18377 df-oposet 38041 df-covers 38131 df-ats 38132 |
This theorem is referenced by: leat3 38160 tendoex 39841 |
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