Theorem frgrwopreglem4 41486

Description: Lemma 4 for frgrwopreg 41488. In a friendship graph each vertex with degree K is connected with a vertex with degree other than K. This corresponds to statement 4 in [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B.". (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.)

Hypotheses

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)
frgrwopreg.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
frgrwopreglem4	⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑎, 𝑏} ∈ 𝐸)

Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝐴,𝑏   𝐺,𝑎,𝑏,𝑥

Allowed substitution hints:   𝐴(𝑎)   𝐵(𝑥,𝑎,𝑏)   𝐷(𝑎,𝑏)   𝐸(𝑥,𝑎,𝑏)   𝐾(𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem frgrwopreglem4

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrwopreg.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		frgrwopreg.d	. . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺)
3		frgrwopreg.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
4		frgrwopreg.b	. . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
5	1, 2, 3, 4	frgrwopreglem3 41485	. . 3 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐷‘𝑎) ≠ (𝐷‘𝑏))
6	1, 2	frgrncvvdeq 41482	. . . 4 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
7		elrabi 3327	. . . . . . . . 9 ⊢ (𝑎 ∈ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → 𝑎 ∈ 𝑉)
8	7, 3	eleq2s 2705	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝑉)
9		sneq 4134	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎})
10	9	difeq2d 3689	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑎}))
11		oveq2 6535	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑎))
12		neleq2 2888	. . . . . . . . . . . 12 ⊢ ((𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑎) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑎)))
13	11, 12	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑎)))
14		fveq2 6088	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐷‘𝑥) = (𝐷‘𝑎))
15	14	eqeq1d 2611	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝐷‘𝑥) = (𝐷‘𝑦) ↔ (𝐷‘𝑎) = (𝐷‘𝑦)))
16	13, 15	imbi12d 332	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑦))))
17	10, 16	raleqbidv 3128	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) ↔ ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑦))))
18	17	rspcv 3277	. . . . . . . 8 ⊢ (𝑎 ∈ 𝑉 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑦))))
19	8, 18	syl 17	. . . . . . 7 ⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑦))))
20	19	adantr 479	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑦))))
21	4	eleq2i 2679	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ (𝑉 ∖ 𝐴))
22		eldif 3549	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝑉 ∖ 𝐴) ↔ (𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴))
23	21, 22	bitri 262	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴))
24		simpll 785	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑏 ∈ 𝑉)
25		eleq1a 2682	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝐴 → (𝑏 = 𝑎 → 𝑏 ∈ 𝐴))
26	25	con3rr3 149	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑏 ∈ 𝐴 → (𝑎 ∈ 𝐴 → ¬ 𝑏 = 𝑎))
27	26	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) → (𝑎 ∈ 𝐴 → ¬ 𝑏 = 𝑎))
28	27	imp 443	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) ∧ 𝑎 ∈ 𝐴) → ¬ 𝑏 = 𝑎)
29		velsn 4140	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ {𝑎} ↔ 𝑏 = 𝑎)
30	28, 29	sylnibr 317	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) ∧ 𝑎 ∈ 𝐴) → ¬ 𝑏 ∈ {𝑎})
31	24, 30	eldifd 3550	. . . . . . . . . 10 ⊢ (((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑏 ∈ (𝑉 ∖ {𝑎}))
32	31	ex 448	. . . . . . . . 9 ⊢ ((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) → (𝑎 ∈ 𝐴 → 𝑏 ∈ (𝑉 ∖ {𝑎})))
33	23, 32	sylbi 205	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 → (𝑎 ∈ 𝐴 → 𝑏 ∈ (𝑉 ∖ {𝑎})))
34	33	impcom 444	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ (𝑉 ∖ {𝑎}))
35		neleq1 2887	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 ∉ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∉ (𝐺 NeighbVtx 𝑎)))
36		fveq2 6088	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (𝐷‘𝑦) = (𝐷‘𝑏))
37	36	eqeq2d 2619	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → ((𝐷‘𝑎) = (𝐷‘𝑦) ↔ (𝐷‘𝑎) = (𝐷‘𝑏)))
38	35, 37	imbi12d 332	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑦)) ↔ (𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑏))))
39	38	rspcv 3277	. . . . . . 7 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → (∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑦)) → (𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑏))))
40	34, 39	syl 17	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑦)) → (𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑏))))
41		nnel 2891	. . . . . . . . 9 ⊢ (¬ 𝑏 ∉ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
42		frgrusgr 41434	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
43		frgrwopreg.e	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (Edg‘𝐺)
44	43	nbusgreledg 40577	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑏, 𝑎} ∈ 𝐸))
45	42, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ FriendGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑏, 𝑎} ∈ 𝐸))
46		prcom 4210	. . . . . . . . . . . . . . 15 ⊢ {𝑏, 𝑎} = {𝑎, 𝑏}
47	46	eleq1i 2678	. . . . . . . . . . . . . 14 ⊢ ({𝑏, 𝑎} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸)
48	45, 47	syl6bb 274	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ FriendGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑎, 𝑏} ∈ 𝐸))
49	48	biimpa 499	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)) → {𝑎, 𝑏} ∈ 𝐸)
50	49	a1d 25	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)) → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸))
51	50	expcom 449	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
52	51	a1d 25	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
53	41, 52	sylbi 205	. . . . . . . 8 ⊢ (¬ 𝑏 ∉ (𝐺 NeighbVtx 𝑎) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
54		eqneqall 2792	. . . . . . . . 9 ⊢ ((𝐷‘𝑎) = (𝐷‘𝑏) → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸))
55	54	2a1d 26	. . . . . . . 8 ⊢ ((𝐷‘𝑎) = (𝐷‘𝑏) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
56	53, 55	ja 171	. . . . . . 7 ⊢ ((𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑏)) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
57	56	com12 32	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑏)) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
58	20, 40, 57	3syld 57	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
59	58	com3l 86	. . . 4 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) → (𝐺 ∈ FriendGraph → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
60	6, 59	mpcom 37	. . 3 ⊢ (𝐺 ∈ FriendGraph → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
61	5, 60	mpdi 43	. 2 ⊢ (𝐺 ∈ FriendGraph → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → {𝑎, 𝑏} ∈ 𝐸))
62	61	ralrimivv 2952	1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑎, 𝑏} ∈ 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 194   ∧ wa 382   = wceq 1474   ∈ wcel 1976   ≠ wne 2779   ∉ wnel 2780  ∀wral 2895  {crab 2899   ∖ cdif 3536  {csn 4124  {cpr 4126  ‘cfv 5790  (class class class)co 6527  Vtxcvtx 40231  Edgcedga 40353   USGraph cusgr 40381   NeighbVtx cnbgr 40552  VtxDegcvtxdg 40683   FriendGraph cfrgr 41430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-xadd 11779  df-fz 12153  df-hash 12935  df-xnn0 40200  df-uhgr 40282  df-ushgr 40283  df-upgr 40310  df-umgr 40311  df-edga 40354  df-uspgr 40382  df-usgr 40383  df-nbgr 40556  df-vtxdg 40684  df-frgr 41431

This theorem is referenced by:  frgrwopreglem5  41487  frgrwopreg1  41489  frgrwopreg2  41490