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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 32064 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 38649 | . . 3 β’ (πΎ β π· β π΄ = {π₯ β π΅ β£ 0 πΆπ₯}) |
6 | 5 | eleq2d 2811 | . 2 β’ (πΎ β π· β (π β π΄ β π β {π₯ β π΅ β£ 0 πΆπ₯})) |
7 | breq2 5143 | . . 3 β’ (π₯ = π β ( 0 πΆπ₯ β 0 πΆπ)) | |
8 | 7 | elrab 3676 | . 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 1533 β wcel 2098 {crab 3424 class class class wbr 5139 βcfv 6534 Basecbs 17145 0.cp0 18380 β ccvr 38626 Atomscatm 38627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-ats 38631 |
This theorem is referenced by: isat2 38651 atcvr0 38652 atbase 38653 isat3 38671 1cvrco 38837 1cvrjat 38840 ltrnatb 39502 |
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