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Mirrors > Home > MPE Home > Th. List > frgrwopreglem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for frgrwopreg 28096: 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 6679 | . 2 ⊢ 𝑉 ∈ V |
3 | frgrwopreg.a | . . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} | |
4 | rabexg 5227 | . . . 4 ⊢ (𝑉 ∈ V → {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ∈ V) | |
5 | 3, 4 | eqeltrid 2917 | . . 3 ⊢ (𝑉 ∈ V → 𝐴 ∈ V) |
6 | frgrwopreg.b | . . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴) | |
7 | difexg 5224 | . . . 4 ⊢ (𝑉 ∈ V → (𝑉 ∖ 𝐴) ∈ V) | |
8 | 6, 7 | eqeltrid 2917 | . . 3 ⊢ (𝑉 ∈ V → 𝐵 ∈ V) |
9 | 5, 8 | jca 514 | . 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 398 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3495 ∖ cdif 3933 ‘cfv 6350 Vtxcvtx 26775 VtxDegcvtxdg 27241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-sn 4562 df-pr 4564 df-uni 4833 df-iota 6309 df-fv 6358 |
This theorem is referenced by: frgrwopreg2 28092 frgrwopreglem5 28094 frgrwopreglem5ALT 28095 frgrwopreg 28096 |
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