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Mirrors > Home > MPE Home > Th. List > frgrwopreglem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for frgrwopreg 30251: the classes 𝐴 and 𝐵 are sets. The definition of 𝐴 and 𝐵 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 | ⊢ 𝑉 = (Vtx‘𝐺) |
frgrwopreg.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
frgrwopreg.a | ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} |
frgrwopreg.b | ⊢ 𝐵 = (𝑉 ∖ 𝐴) |
Ref | Expression |
---|---|
frgrwopreglem1 | ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrwopreg.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | fvexi 6905 | . 2 ⊢ 𝑉 ∈ V |
3 | frgrwopreg.a | . . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} | |
4 | rabexg 5329 | . . . 4 ⊢ (𝑉 ∈ V → {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ∈ V) | |
5 | 3, 4 | eqeltrid 2830 | . . 3 ⊢ (𝑉 ∈ V → 𝐴 ∈ V) |
6 | frgrwopreg.b | . . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴) | |
7 | difexg 5325 | . . . 4 ⊢ (𝑉 ∈ V → (𝑉 ∖ 𝐴) ∈ V) | |
8 | 6, 7 | eqeltrid 2830 | . . 3 ⊢ (𝑉 ∈ V → 𝐵 ∈ V) |
9 | 5, 8 | jca 510 | . 2 ⊢ (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
10 | 2, 9 | ax-mp 5 | 1 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 ∈ wcel 2099 {crab 3420 Vcvv 3463 ∖ cdif 3944 ‘cfv 6544 Vtxcvtx 28927 VtxDegcvtxdg 29397 |
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 2697 ax-sep 5295 ax-nul 5302 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-sn 4625 df-pr 4627 df-uni 4907 df-iota 6496 df-fv 6552 |
This theorem is referenced by: frgrwopreg2 30247 frgrwopreglem5 30249 frgrwopreglem5ALT 30250 frgrwopreg 30251 |
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