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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 31323 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 37793 | . . 3 β’ (πΎ β π· β π΄ = {π₯ β π΅ β£ 0 πΆπ₯}) |
6 | 5 | eleq2d 2820 | . 2 β’ (πΎ β π· β (π β π΄ β π β {π₯ β π΅ β£ 0 πΆπ₯})) |
7 | breq2 5110 | . . 3 β’ (π₯ = π β ( 0 πΆπ₯ β 0 πΆπ)) | |
8 | 7 | elrab 3646 | . 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 397 = wceq 1542 β wcel 2107 {crab 3406 class class class wbr 5106 βcfv 6497 Basecbs 17088 0.cp0 18317 β ccvr 37770 Atomscatm 37771 |
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-sep 5257 ax-nul 5264 ax-pr 5385 |
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-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 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-iota 6449 df-fun 6499 df-fv 6505 df-ats 37775 |
This theorem is referenced by: isat2 37795 atcvr0 37796 atbase 37797 isat3 37815 1cvrco 37981 1cvrjat 37984 ltrnatb 38646 |
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