![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > atcvr0 | Structured version Visualization version GIF version |
Description: An atom covers zero. (atcv0 31326 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 2737 | . . 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 37777 | . 2 β’ (πΎ β π· β (π β π΄ β (π β (BaseβπΎ) β§ 0 πΆπ))) |
6 | 5 | simplbda 501 | 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 397 = wceq 1542 β wcel 2107 class class class wbr 5110 βcfv 6501 Basecbs 17090 0.cp0 18319 β ccvr 37753 Atomscatm 37754 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-ats 37758 |
This theorem is referenced by: 0ltat 37782 leatb 37783 atnle0 37800 atlen0 37801 atcmp 37802 atcvreq0 37805 atcvr0eq 37918 lnnat 37919 athgt 37948 ps-2 37970 lhp0lt 38495 |
Copyright terms: Public domain | W3C validator |