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Mirrors > Home > MPE Home > Th. List > frgrwopreglem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for frgrwopreg 30120: 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 6905 | . 2 β’ π β V |
3 | frgrwopreg.a | . . . 4 β’ π΄ = {π₯ β π β£ (π·βπ₯) = πΎ} | |
4 | rabexg 5327 | . . . 4 β’ (π β V β {π₯ β π β£ (π·βπ₯) = πΎ} β V) | |
5 | 3, 4 | eqeltrid 2832 | . . 3 β’ (π β V β π΄ β V) |
6 | frgrwopreg.b | . . . 4 β’ π΅ = (π β π΄) | |
7 | difexg 5323 | . . . 4 β’ (π β V β (π β π΄) β V) | |
8 | 6, 7 | eqeltrid 2832 | . . 3 β’ (π β V β π΅ β V) |
9 | 5, 8 | jca 511 | . 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 395 = wceq 1534 β wcel 2099 {crab 3427 Vcvv 3469 β cdif 3941 βcfv 6542 Vtxcvtx 28796 VtxDegcvtxdg 29266 |
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-ext 2698 ax-sep 5293 ax-nul 5300 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-sn 4625 df-pr 4627 df-uni 4904 df-iota 6494 df-fv 6550 |
This theorem is referenced by: frgrwopreg2 30116 frgrwopreglem5 30118 frgrwopreglem5ALT 30119 frgrwopreg 30120 |
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