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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 32143 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 38752 | . . 3 β’ (πΎ β π· β π΄ = {π₯ β π΅ β£ 0 πΆπ₯}) |
6 | 5 | eleq2d 2815 | . 2 β’ (πΎ β π· β (π β π΄ β π β {π₯ β π΅ β£ 0 πΆπ₯})) |
7 | breq2 5147 | . . 3 β’ (π₯ = π β ( 0 πΆπ₯ β 0 πΆπ)) | |
8 | 7 | elrab 3681 | . 2 β’ (π β {π₯ β π΅ β£ 0 πΆπ₯} β (π β π΅ β§ 0 πΆπ)) |
9 | 6, 8 | bitrdi 287 | 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 {crab 3428 class class class wbr 5143 βcfv 6543 Basecbs 17174 0.cp0 18409 β ccvr 38729 Atomscatm 38730 |
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 5294 ax-nul 5301 ax-pr 5424 |
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 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-ats 38734 |
This theorem is referenced by: isat2 38754 atcvr0 38755 atbase 38756 isat3 38774 1cvrco 38940 1cvrjat 38943 ltrnatb 39605 |
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