Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > frgrawopreglem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for frgrawopreg 26576. In a friendship graph, 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.) |
Ref | Expression |
---|---|
frgrawopreg.a | ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} |
frgrawopreg.b | ⊢ 𝐵 = (𝑉 ∖ 𝐴) |
Ref | Expression |
---|---|
frgrawopreglem1 | ⊢ (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frisusgra 26519 | . 2 ⊢ (𝑉 FriendGrph 𝐸 → 𝑉 USGrph 𝐸) | |
2 | usgrav 25867 | . 2 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) | |
3 | frgrawopreg.a | . . . . 5 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} | |
4 | rabexg 4739 | . . . . 5 ⊢ (𝑉 ∈ V → {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} ∈ V) | |
5 | 3, 4 | syl5eqel 2692 | . . . 4 ⊢ (𝑉 ∈ V → 𝐴 ∈ V) |
6 | frgrawopreg.b | . . . . 5 ⊢ 𝐵 = (𝑉 ∖ 𝐴) | |
7 | difexg 4735 | . . . . 5 ⊢ (𝑉 ∈ V → (𝑉 ∖ 𝐴) ∈ V) | |
8 | 6, 7 | syl5eqel 2692 | . . . 4 ⊢ (𝑉 ∈ V → 𝐵 ∈ V) |
9 | 5, 8 | jca 553 | . . 3 ⊢ (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
10 | 9 | adantr 480 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
11 | 1, 2, 10 | 3syl 18 | 1 ⊢ (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∖ cdif 3537 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 USGrph cusg 25859 VDeg cvdg 26420 FriendGrph cfrgra 26515 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-usgra 25862 df-frgra 26516 |
This theorem is referenced by: frgrawopreglem5 26575 frgrawopreg 26576 frgrawopreg1 26577 frgrawopreg2 26578 |
Copyright terms: Public domain | W3C validator |