Theorem frgrwopreg 30379

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 30368	. 2 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V)
6		hashv01gt1 14270	. . . 4 ⊢ (𝐴 ∈ V → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
7		hasheq0 14288	. . . . . 6 ⊢ (𝐴 ∈ V → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
8		biidd 262	. . . . . 6 ⊢ (𝐴 ∈ V → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = 1))
9		biidd 262	. . . . . 6 ⊢ (𝐴 ∈ V → (1 < (♯‘𝐴) ↔ 1 < (♯‘𝐴)))
10	7, 8, 9	3orbi123d 1438	. . . . 5 ⊢ (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) ↔ (𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴))))
11		hashv01gt1 14270	. . . . . . 7 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)))
12		hasheq0 14288	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
13		biidd 262	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ (♯‘𝐵) = 1))
14		biidd 262	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (1 < (♯‘𝐵) ↔ 1 < (♯‘𝐵)))
15	12, 13, 14	3orbi123d 1438	. . . . . . . 8 ⊢ (𝐵 ∈ V → (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) ↔ (𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))))
16		olc 869	. . . . . . . . . . 11 ⊢ (𝐵 = ∅ → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
17	16	olcd 875	. . . . . . . . . 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 868	. . . . . . . . . . 11 ⊢ ((♯‘𝐵) = 1 → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
20	19	olcd 875	. . . . . . . . . 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 869	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
23	22	orcd 874	. . . . . . . . . . . 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 868	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) = 1 → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
26	25	orcd 874	. . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
29	1, 2, 3, 4, 28	frgrwopreglem5 30377	. . . . . . . . . . . . . 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 30317	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
31		simplll 775	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝐺 ∈ USGraph)
32		elrabi 3641	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∈ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → 𝑎 ∈ 𝑉)
33	32, 3	eleq2s 2853	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝑉)
34	33	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑎 ∈ 𝑉)
35	34	ad3antlr 732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑎 ∈ 𝑉)
36		rabidim1 3420	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → 𝑥 ∈ 𝑉)
37	36, 3	eleq2s 2853	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑉)
38	37	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑉)
39	38	ad3antlr 732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑥 ∈ 𝑉)
40		simprl 771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑎 ≠ 𝑥)
41		eldifi 4082	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 ∈ (𝑉 ∖ 𝐴) → 𝑏 ∈ 𝑉)
42	41, 4	eleq2s 2853	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 ∈ 𝐵 → 𝑏 ∈ 𝑉)
43	42	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑏 ∈ 𝑉)
44	43	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑏 ∈ 𝑉)
45		eldifi 4082	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (𝑉 ∖ 𝐴) → 𝑦 ∈ 𝑉)
46	45, 4	eleq2s 2853	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑉)
47	46	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝑉)
48	47	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑦 ∈ 𝑉)
49		simprr 773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑏 ≠ 𝑦)
50	1, 28	4cyclusnfrgr 30348	. . . . . . . . . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . . . . . . . . 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 430	. . . . . . . . . . . . . . . . . . . . 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 1350	. . . . . . . . . . . . . . . . . . . 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 3036	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
55		pm2.21 123	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
56	54, 55	sylbi 217	. . . . . . . . . . . . . . . . . . . 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 3192	. . . . . . . . . . . . . . . . . 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 3192	. . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . 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 1120	. . . . . . . . . . . 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 1431	. . . . . . . . . 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 1431	. . . . . . . 8 ⊢ ((𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
69	15, 68	biimtrdi 253	. . . . . . 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 253	. . . 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 406	. 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 395   ∨ wo 848   ∨ w3o 1086   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2931   ∉ wnel 3035  ∃wrex 3059  {crab 3398  Vcvv 3439   ∖ cdif 3897  ∅c0 4284  {cpr 4581   class class class wbr 5097  ‘cfv 6491  0cc0 11028  1c1 11029   < clt 11168  ♯chash 14255  Vtxcvtx 29050  Edgcedg 29101  USGraphcusgr 29203  VtxDegcvtxdg 29520   FriendGraph cfrgr 30314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-n0 12404  df-xnn0 12477  df-z 12491  df-uz 12754  df-xadd 13029  df-fz 13426  df-hash 14256  df-edg 29102  df-uhgr 29112  df-ushgr 29113  df-upgr 29136  df-umgr 29137  df-uspgr 29204  df-usgr 29205  df-nbgr 29387  df-vtxdg 29521  df-frgr 30315

This theorem is referenced by:  frgrregorufr0  30380