![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > isat2 | Structured version Visualization version GIF version |
Description: The predicate "is an atom". (elatcv0 32144 analog.) (Contributed by NM, 18-Jun-2012.) |
Ref | Expression |
---|---|
isatom.b | β’ π΅ = (BaseβπΎ) |
isatom.z | β’ 0 = (0.βπΎ) |
isatom.c | β’ πΆ = ( β βπΎ) |
isatom.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
isat2 | β’ ((πΎ β π· β§ π β π΅) β (π β π΄ β 0 πΆπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isatom.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | isatom.z | . . 3 β’ 0 = (0.βπΎ) | |
3 | isatom.c | . . 3 β’ πΆ = ( β βπΎ) | |
4 | isatom.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
5 | 1, 2, 3, 4 | isat 38752 | . 2 β’ (πΎ β π· β (π β π΄ β (π β π΅ β§ 0 πΆπ))) |
6 | 5 | baibd 539 | 1 β’ ((πΎ β π· β§ π β π΅) β (π β π΄ β 0 πΆπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 class class class wbr 5142 βcfv 6542 Basecbs 17173 0.cp0 18408 β ccvr 38728 Atomscatm 38729 |
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 5293 ax-nul 5300 ax-pr 5423 |
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 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ats 38733 |
This theorem is referenced by: llncvrlpln 39025 lplncvrlvol 39083 lhpm0atN 39496 |
Copyright terms: Public domain | W3C validator |