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Mirrors > Home > MPE Home > Th. List > Mathboxes > frgrwopreglem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for frgrwopreg 41486: 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 | fvex 6113 | . . 3 ⊢ (Vtx‘𝐺) ∈ V | |
3 | 1, 2 | eqeltri 2684 | . 2 ⊢ 𝑉 ∈ V |
4 | frgrwopreg.a | . . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} | |
5 | rabexg 4739 | . . . 4 ⊢ (𝑉 ∈ V → {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ∈ V) | |
6 | 4, 5 | syl5eqel 2692 | . . 3 ⊢ (𝑉 ∈ V → 𝐴 ∈ V) |
7 | frgrwopreg.b | . . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴) | |
8 | difexg 4735 | . . . 4 ⊢ (𝑉 ∈ V → (𝑉 ∖ 𝐴) ∈ V) | |
9 | 7, 8 | syl5eqel 2692 | . . 3 ⊢ (𝑉 ∈ V → 𝐵 ∈ V) |
10 | 6, 9 | jca 553 | . 2 ⊢ (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
11 | 3, 10 | ax-mp 5 | 1 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∖ cdif 3537 ‘cfv 5804 Vtxcvtx 25673 VtxDegcvtxdg 40681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 df-fv 5812 |
This theorem is referenced by: frgrwopreglem5 41485 frgrwopreg 41486 frgrwopreg2 41488 |
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