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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atcvr0 | Structured version Visualization version GIF version |
Description: An atom covers zero. (atcv0 32165 analog.) (Contributed by NM, 4-Nov-2011.) |
Ref | Expression |
---|---|
atomcvr0.z | β’ 0 = (0.βπΎ) |
atomcvr0.c | β’ πΆ = ( β βπΎ) |
atomcvr0.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
atcvr0 | β’ ((πΎ β π· β§ π β π΄) β 0 πΆπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | atomcvr0.z | . . 3 β’ 0 = (0.βπΎ) | |
3 | atomcvr0.c | . . 3 β’ πΆ = ( β βπΎ) | |
4 | atomcvr0.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
5 | 1, 2, 3, 4 | isat 38758 | . 2 β’ (πΎ β π· β (π β π΄ β (π β (BaseβπΎ) β§ 0 πΆπ))) |
6 | 5 | simplbda 499 | 1 β’ ((πΎ β π· β§ π β π΄) β 0 πΆπ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 Basecbs 17180 0.cp0 18415 β ccvr 38734 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-sep 5299 ax-nul 5306 ax-pr 5429 |
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-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 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-iota 6500 df-fun 6550 df-fv 6556 df-ats 38739 |
This theorem is referenced by: 0ltat 38763 leatb 38764 atnle0 38781 atlen0 38782 atcmp 38783 atcvreq0 38786 atcvr0eq 38899 lnnat 38900 athgt 38929 ps-2 38951 lhp0lt 39476 |
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