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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 38656 | . 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 476 | 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 395 β¨ wo 844 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5139 βcfv 6534 Basecbs 17145 lecple 17205 0.cp0 18380 OPcops 38536 Atomscatm 38627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-proset 18252 df-poset 18270 df-plt 18287 df-glb 18304 df-p0 18382 df-oposet 38540 df-covers 38630 df-ats 38631 |
This theorem is referenced by: leat3 38659 tendoex 40340 |
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