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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atn0 | Structured version Visualization version GIF version |
Description: An atom is not zero. (atne0 31329 analog.) (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
atne0.z | β’ 0 = (0.βπΎ) |
atne0.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
atn0 | β’ ((πΎ β AtLat β§ π β π΄) β π β 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | eqid 2733 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
3 | atne0.z | . . . 4 β’ 0 = (0.βπΎ) | |
4 | atne0.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
5 | 1, 2, 3, 4 | isat3 37815 | . . 3 β’ (πΎ β AtLat β (π β π΄ β (π β (BaseβπΎ) β§ π β 0 β§ βπ₯ β (BaseβπΎ)(π₯(leβπΎ)π β (π₯ = π β¨ π₯ = 0 ))))) |
6 | simp2 1138 | . . 3 β’ ((π β (BaseβπΎ) β§ π β 0 β§ βπ₯ β (BaseβπΎ)(π₯(leβπΎ)π β (π₯ = π β¨ π₯ = 0 ))) β π β 0 ) | |
7 | 5, 6 | syl6bi 253 | . 2 β’ (πΎ β AtLat β (π β π΄ β π β 0 )) |
8 | 7 | imp 408 | 1 β’ ((πΎ β AtLat β§ π β π΄) β π β 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β¨ wo 846 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 class class class wbr 5106 βcfv 6497 Basecbs 17088 lecple 17145 0.cp0 18317 Atomscatm 37771 AtLatcal 37772 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-plt 18224 df-glb 18241 df-p0 18319 df-covers 37774 df-ats 37775 df-atl 37806 |
This theorem is referenced by: atncvrN 37823 atnle 37825 atlatmstc 37827 intnatN 37916 atcvrneN 37939 atcvrj2b 37941 2llnm3N 38078 pmapjat1 38362 lhpocnle 38525 lhpmatb 38540 lhp2atnle 38542 trlatn0 38681 ltrnnidn 38683 trlnidatb 38686 cdlemg33c 39217 cdlemg33e 39219 dihatexv 39847 |
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