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Mirrors > Home > MPE Home > Th. List > frgrwopreglem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for frgrwopreg 28378: 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 6720 | . 2 ⊢ 𝑉 ∈ V |
3 | frgrwopreg.a | . . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} | |
4 | rabexg 5213 | . . . 4 ⊢ (𝑉 ∈ V → {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ∈ V) | |
5 | 3, 4 | eqeltrid 2838 | . . 3 ⊢ (𝑉 ∈ V → 𝐴 ∈ V) |
6 | frgrwopreg.b | . . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴) | |
7 | difexg 5209 | . . . 4 ⊢ (𝑉 ∈ V → (𝑉 ∖ 𝐴) ∈ V) | |
8 | 6, 7 | eqeltrid 2838 | . . 3 ⊢ (𝑉 ∈ V → 𝐵 ∈ V) |
9 | 5, 8 | jca 515 | . 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 399 = wceq 1543 ∈ wcel 2110 {crab 3058 Vcvv 3401 ∖ cdif 3854 ‘cfv 6369 Vtxcvtx 27059 VtxDegcvtxdg 27525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-sn 4532 df-pr 4534 df-uni 4810 df-iota 6327 df-fv 6377 |
This theorem is referenced by: frgrwopreg2 28374 frgrwopreglem5 28376 frgrwopreglem5ALT 28377 frgrwopreg 28378 |
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