Theorem frgrwopreg 30415

Description: In a friendship graph there are either no vertices (𝐴 = ∅) or exactly one vertex ((♯‘𝐴) = 1) having degree 𝐾, or all (𝐵 = ∅) or all except one vertices ((♯‘𝐵) = 1) have degree 𝐾. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 3-Jan-2022.)

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)

Assertion

Ref	Expression
frgrwopreg	⊢ (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))

Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝑥,𝐵

Proof of Theorem frgrwopreg

Dummy variables 𝑎 𝑏 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrwopreg.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		frgrwopreg.d	. . 3 ⊢ 𝐷 = (VtxDeg‘𝐺)
3		frgrwopreg.a	. . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
4		frgrwopreg.b	. . 3 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
5	1, 2, 3, 4	frgrwopreglem1 30404	. 2 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V)
6		hashv01gt1 14302	. . . 4 ⊢ (𝐴 ∈ V → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
7		hasheq0 14320	. . . . . 6 ⊢ (𝐴 ∈ V → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
8		biidd 264	. . . . . 6 ⊢ (𝐴 ∈ V → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = 1))
9		biidd 264	. . . . . 6 ⊢ (𝐴 ∈ V → (1 < (♯‘𝐴) ↔ 1 < (♯‘𝐴)))
10	7, 8, 9	3orbi123d 1444	. . . . 5 ⊢ (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) ↔ (𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴))))
11		hashv01gt1 14302	. . . . . . 7 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)))
12		hasheq0 14320	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
13		biidd 264	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ (♯‘𝐵) = 1))
14		biidd 264	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (1 < (♯‘𝐵) ↔ 1 < (♯‘𝐵)))
15	12, 13, 14	3orbi123d 1444	. . . . . . . 8 ⊢ (𝐵 ∈ V → (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) ↔ (𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))))
16		olc 875	. . . . . . . . . . 11 ⊢ (𝐵 = ∅ → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
17	16	olcd 881	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
18	17	2a1d 26	. . . . . . . . 9 ⊢ (𝐵 = ∅ → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
19		orc 874	. . . . . . . . . . 11 ⊢ ((♯‘𝐵) = 1 → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
20	19	olcd 881	. . . . . . . . . 10 ⊢ ((♯‘𝐵) = 1 → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
21	20	2a1d 26	. . . . . . . . 9 ⊢ ((♯‘𝐵) = 1 → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
22		olc 875	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
23	22	orcd 880	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
24	23	2a1d 26	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
25		orc 874	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) = 1 → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
26	25	orcd 880	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) = 1 → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
27	26	2a1d 26	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) = 1 → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
28		eqid 2741	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
29	1, 2, 3, 4, 28	frgrwopreglem5 30413	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))))
30		frgrusgr 30353	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
31		simplll 781	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝐺 ∈ USGraph)
32		elrabi 3627	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∈ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → 𝑎 ∈ 𝑉)
33	32, 3	eleq2s 2859	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝑉)
34	33	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑎 ∈ 𝑉)
35	34	ad3antlr 738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑎 ∈ 𝑉)
36		rabidim1 3415	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → 𝑥 ∈ 𝑉)
37	36, 3	eleq2s 2859	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑉)
38	37	adantl 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑉)
39	38	ad3antlr 738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑥 ∈ 𝑉)
40		simprl 777	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑎 ≠ 𝑥)
41		eldifi 4064	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 ∈ (𝑉 ∖ 𝐴) → 𝑏 ∈ 𝑉)
42	41, 4	eleq2s 2859	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 ∈ 𝐵 → 𝑏 ∈ 𝑉)
43	42	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑏 ∈ 𝑉)
44	43	ad2antlr 734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑏 ∈ 𝑉)
45		eldifi 4064	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (𝑉 ∖ 𝐴) → 𝑦 ∈ 𝑉)
46	45, 4	eleq2s 2859	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑉)
47	46	adantl 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝑉)
48	47	ad2antlr 734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑦 ∈ 𝑉)
49		simprr 779	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑏 ≠ 𝑦)
50	1, 28	4cyclusnfrgr 30384	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑎 ≠ 𝑥) ∧ (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑏 ≠ 𝑦)) → ((({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → 𝐺 ∉ FriendGraph ))
51	31, 35, 39, 40, 44, 48, 49, 50	syl133anc 1402	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → ((({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → 𝐺 ∉ FriendGraph ))
52	51	exp4b 432	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) → (({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺)) → 𝐺 ∉ FriendGraph ))))
53	52	3impd 1356	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → 𝐺 ∉ FriendGraph ))
54		df-nel 3041	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
55		pm2.21 123	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
56	54, 55	sylbi 219	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 ∉ FriendGraph → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
57	53, 56	syl6 35	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
58	57	rexlimdvva 3198	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
59	58	rexlimdvva 3198	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
60	59	com23 86	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
61	30, 60	mpcom 38	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
62	61	3ad2ant1 1140	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
63	29, 62	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
64	63	3exp 1126	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FriendGraph → (1 < (♯‘𝐴) → (1 < (♯‘𝐵) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
65	64	com3l 89	. . . . . . . . . . 11 ⊢ (1 < (♯‘𝐴) → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
66	24, 27, 65	3jaoi 1437	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
67	66	com12 32	. . . . . . . . 9 ⊢ (1 < (♯‘𝐵) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
68	18, 21, 67	3jaoi 1437	. . . . . . . 8 ⊢ ((𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
69	15, 68	biimtrdi 255	. . . . . . 7 ⊢ (𝐵 ∈ V → (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))))
70	11, 69	mpd 15	. . . . . 6 ⊢ (𝐵 ∈ V → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
71	70	com12 32	. . . . 5 ⊢ ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐵 ∈ V → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
72	10, 71	biimtrdi 255	. . . 4 ⊢ (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐵 ∈ V → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))))
73	6, 72	mpd 15	. . 3 ⊢ (𝐴 ∈ V → (𝐵 ∈ V → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
74	73	imp 408	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
75	5, 74	ax-mp 5	1 ⊢ (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 854   ∨ w3o 1092   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ≠ wne 2936   ∉ wnel 3040  ∃wrex 3065  {crab 3393  Vcvv 3433   ∖ cdif 3882  ∅c0 4264  {cpr 4560   class class class wbr 5075  ‘cfv 6489  0cc0 11033  1c1 11034   < clt 11174  ♯chash 14287  Vtxcvtx 29087  Edgcedg 29138  USGraphcusgr 29240  VtxDegcvtxdg 29556   FriendGraph cfrgr 30350

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-n0 12433  df-xnn0 12506  df-z 12520  df-uz 12784  df-xadd 13059  df-fz 13457  df-hash 14288  df-edg 29139  df-uhgr 29149  df-ushgr 29150  df-upgr 29173  df-umgr 29174  df-uspgr 29241  df-usgr 29242  df-nbgr 29424  df-vtxdg 29557  df-frgr 30351

This theorem is referenced by:  frgrregorufr0  30416