Theorem frgrwopreglem4 41484

Description: Lemma 4 for frgrwopreg 41486. 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 41483	. . 3 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐷‘𝑎) ≠ (𝐷‘𝑏))
6	1, 2	frgrncvvdeq 41480	. . . 4 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
7		elrabi 3328	. . . . . . . . 9 ⊢ (𝑎 ∈ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → 𝑎 ∈ 𝑉)
8	7, 3	eleq2s 2706	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝑉)
9		sneq 4135	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎})
10	9	difeq2d 3690	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑎}))
11		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑎))
12		neleq2 2889	. . . . . . . . . . . 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 6103	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐷‘𝑥) = (𝐷‘𝑎))
15	14	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝐷‘𝑥) = (𝐷‘𝑦) ↔ (𝐷‘𝑎) = (𝐷‘𝑦)))
16	13, 15	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑦))))
17	10, 16	raleqbidv 3129	. . . . . . . . 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 3278	. . . . . . . 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 480	. . . . . 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 2680	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ (𝑉 ∖ 𝐴))
22		eldif 3550	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝑉 ∖ 𝐴) ↔ (𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴))
23	21, 22	bitri 263	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴))
24		simpll 786	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑏 ∈ 𝑉)
25		eleq1a 2683	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝐴 → (𝑏 = 𝑎 → 𝑏 ∈ 𝐴))
26	25	con3rr3 150	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑏 ∈ 𝐴 → (𝑎 ∈ 𝐴 → ¬ 𝑏 = 𝑎))
27	26	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) → (𝑎 ∈ 𝐴 → ¬ 𝑏 = 𝑎))
28	27	imp 444	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) ∧ 𝑎 ∈ 𝐴) → ¬ 𝑏 = 𝑎)
29		velsn 4141	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ {𝑎} ↔ 𝑏 = 𝑎)
30	28, 29	sylnibr 318	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) ∧ 𝑎 ∈ 𝐴) → ¬ 𝑏 ∈ {𝑎})
31	24, 30	eldifd 3551	. . . . . . . . . 10 ⊢ (((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑏 ∈ (𝑉 ∖ {𝑎}))
32	31	ex 449	. . . . . . . . 9 ⊢ ((𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴) → (𝑎 ∈ 𝐴 → 𝑏 ∈ (𝑉 ∖ {𝑎})))
33	23, 32	sylbi 206	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 → (𝑎 ∈ 𝐴 → 𝑏 ∈ (𝑉 ∖ {𝑎})))
34	33	impcom 445	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ (𝑉 ∖ {𝑎}))
35		neleq1 2888	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 ∉ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∉ (𝐺 NeighbVtx 𝑎)))
36		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (𝐷‘𝑦) = (𝐷‘𝑏))
37	36	eqeq2d 2620	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → ((𝐷‘𝑎) = (𝐷‘𝑦) ↔ (𝐷‘𝑎) = (𝐷‘𝑏)))
38	35, 37	imbi12d 333	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑦)) ↔ (𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷‘𝑎) = (𝐷‘𝑏))))
39	38	rspcv 3278	. . . . . . 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 2892	. . . . . . . . 9 ⊢ (¬ 𝑏 ∉ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
42		frgrusgr 41432	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
43		frgrwopreg.e	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (Edg‘𝐺)
44	43	nbusgreledg 40575	. . . . . . . . . . . . . . 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 4211	. . . . . . . . . . . . . . 15 ⊢ {𝑏, 𝑎} = {𝑎, 𝑏}
47	46	eleq1i 2679	. . . . . . . . . . . . . 14 ⊢ ({𝑏, 𝑎} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸)
48	45, 47	syl6bb 275	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ FriendGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑎, 𝑏} ∈ 𝐸))
49	48	biimpa 500	. . . . . . . . . . . 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 450	. . . . . . . . . 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 206	. . . . . . . 8 ⊢ (¬ 𝑏 ∉ (𝐺 NeighbVtx 𝑎) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑎) ≠ (𝐷‘𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
54		eqneqall 2793	. . . . . . . . 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 172	. . . . . . 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 58	. . . . 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 87	. . . 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 44	. 2 ⊢ (𝐺 ∈ FriendGraph → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → {𝑎, 𝑏} ∈ 𝐸))
62	61	ralrimivv 2953	1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑎, 𝑏} ∈ 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  {crab 2900   ∖ cdif 3537  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   NeighbVtx cnbgr 40550  VtxDegcvtxdg 40681   FriendGraph cfrgr 41428

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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-uhgr 25724  df-ushgr 25725  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-nbgr 40554  df-vtxdg 40682  df-frgr 41429

This theorem is referenced by:  frgrwopreglem5  41485  frgrwopreg1  41487  frgrwopreg2  41488