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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > leat3 | Structured version Visualization version GIF version |
Description: A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.) |
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 |
---|---|
leat3 | β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ π β€ π) β (π β 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 | leat 38765 | . 2 β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ π β€ π) β (π = π β¨ π = 0 )) |
6 | simpl3 1191 | . . . 4 β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ π β€ π) β π β π΄) | |
7 | eleq1a 2824 | . . . 4 β’ (π β π΄ β (π = π β π β π΄)) | |
8 | 6, 7 | syl 17 | . . 3 β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ π β€ π) β (π = π β π β π΄)) |
9 | 8 | orim1d 964 | . 2 β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ π β€ π) β ((π = π β¨ π = 0 ) β (π β π΄ β¨ π = 0 ))) |
10 | 5, 9 | mpd 15 | 1 β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ π β€ π) β (π β π΄ β¨ π = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β¨ wo 846 β§ w3a 1085 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 Basecbs 17179 lecple 17239 0.cp0 18414 OPcops 38644 Atomscatm 38735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-proset 18286 df-poset 18304 df-plt 18321 df-glb 18338 df-p0 18416 df-oposet 38648 df-covers 38738 df-ats 38739 |
This theorem is referenced by: cdleme22b 39814 |
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