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Mirrors > Home > MPE Home > Th. List > frgrwopreglem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for frgrwopreg 29309: the classes π΄ and π΅ are sets. The definition of π΄ and π΅ corresponds to definition 3 in [Huneke] p. 2: "Let A be the set of all vertices of degree k, let B be the set of all vertices of degree different from k, ..." (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 10-May-2021.) |
Ref | Expression |
---|---|
frgrwopreg.v | β’ π = (VtxβπΊ) |
frgrwopreg.d | β’ π· = (VtxDegβπΊ) |
frgrwopreg.a | β’ π΄ = {π₯ β π β£ (π·βπ₯) = πΎ} |
frgrwopreg.b | β’ π΅ = (π β π΄) |
Ref | Expression |
---|---|
frgrwopreglem1 | β’ (π΄ β V β§ π΅ β V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrwopreg.v | . . 3 β’ π = (VtxβπΊ) | |
2 | 1 | fvexi 6861 | . 2 β’ π β V |
3 | frgrwopreg.a | . . . 4 β’ π΄ = {π₯ β π β£ (π·βπ₯) = πΎ} | |
4 | rabexg 5293 | . . . 4 β’ (π β V β {π₯ β π β£ (π·βπ₯) = πΎ} β V) | |
5 | 3, 4 | eqeltrid 2842 | . . 3 β’ (π β V β π΄ β V) |
6 | frgrwopreg.b | . . . 4 β’ π΅ = (π β π΄) | |
7 | difexg 5289 | . . . 4 β’ (π β V β (π β π΄) β V) | |
8 | 6, 7 | eqeltrid 2842 | . . 3 β’ (π β V β π΅ β V) |
9 | 5, 8 | jca 513 | . 2 β’ (π β V β (π΄ β V β§ π΅ β V)) |
10 | 2, 9 | ax-mp 5 | 1 β’ (π΄ β V β§ π΅ β V) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 = wceq 1542 β wcel 2107 {crab 3410 Vcvv 3448 β cdif 3912 βcfv 6501 Vtxcvtx 27989 VtxDegcvtxdg 28455 |
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-ext 2708 ax-sep 5261 ax-nul 5268 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-sn 4592 df-pr 4594 df-uni 4871 df-iota 6453 df-fv 6509 |
This theorem is referenced by: frgrwopreg2 29305 frgrwopreglem5 29307 frgrwopreglem5ALT 29308 frgrwopreg 29309 |
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