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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isat | Structured version Visualization version GIF version |
Description: The predicate "is an atom". (ela 31587 analog.) (Contributed by NM, 18-Sep-2011.) |
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 |
---|---|
isat | β’ (πΎ β π· β (π β π΄ β (π β π΅ β§ 0 πΆπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isatom.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | isatom.z | . . . 4 β’ 0 = (0.βπΎ) | |
3 | isatom.c | . . . 4 β’ πΆ = ( β βπΎ) | |
4 | isatom.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
5 | 1, 2, 3, 4 | pats 38150 | . . 3 β’ (πΎ β π· β π΄ = {π₯ β π΅ β£ 0 πΆπ₯}) |
6 | 5 | eleq2d 2819 | . 2 β’ (πΎ β π· β (π β π΄ β π β {π₯ β π΅ β£ 0 πΆπ₯})) |
7 | breq2 5152 | . . 3 β’ (π₯ = π β ( 0 πΆπ₯ β 0 πΆπ)) | |
8 | 7 | elrab 3683 | . 2 β’ (π β {π₯ β π΅ β£ 0 πΆπ₯} β (π β π΅ β§ 0 πΆπ)) |
9 | 6, 8 | bitrdi 286 | 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 396 = wceq 1541 β wcel 2106 {crab 3432 class class class wbr 5148 βcfv 6543 Basecbs 17143 0.cp0 18375 β ccvr 38127 Atomscatm 38128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ats 38132 |
This theorem is referenced by: isat2 38152 atcvr0 38153 atbase 38154 isat3 38172 1cvrco 38338 1cvrjat 38341 ltrnatb 39003 |
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