Theorem frgrawopreg 26576

Description: In a friendship graph there are either no vertices or exactly one vertex having degree K, or all or all except one vertices have degree K. (Contributed by Alexander van der Vekens, 31-Dec-2017.)

Hypotheses

Ref	Expression
frgrawopreg.a	⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b	⊢ 𝐵 = (𝑉 ∖ 𝐴)

Assertion

Ref	Expression
frgrawopreg	⊢ (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉   𝑥,𝐵

Proof of Theorem frgrawopreg

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

Step	Hyp	Ref	Expression
1		frgrawopreg.a	. . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
2		frgrawopreg.b	. . 3 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
3	1, 2	frgrawopreglem1 26571	. 2 ⊢ (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		hashv01gt1 12995	. . . 4 ⊢ (𝐴 ∈ V → ((#‘𝐴) = 0 ∨ (#‘𝐴) = 1 ∨ 1 < (#‘𝐴)))
5		hashv01gt1 12995	. . . 4 ⊢ (𝐵 ∈ V → ((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)))
6	4, 5	anim12i 588	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (((#‘𝐴) = 0 ∨ (#‘𝐴) = 1 ∨ 1 < (#‘𝐴)) ∧ ((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵))))
7		hasheq0 13015	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ V → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
8	7	biimpd 218	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → ((#‘𝐴) = 0 → 𝐴 = ∅))
9	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((#‘𝐴) = 0 → 𝐴 = ∅))
10	9	impcom 445	. . . . . . . . . 10 ⊢ (((#‘𝐴) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → 𝐴 = ∅)
11	10	olcd 407	. . . . . . . . 9 ⊢ (((#‘𝐴) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ((#‘𝐴) = 1 ∨ 𝐴 = ∅))
12	11	orcd 406	. . . . . . . 8 ⊢ (((#‘𝐴) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
13	12	a1d 25	. . . . . . 7 ⊢ (((#‘𝐴) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
14	13	ex 449	. . . . . 6 ⊢ ((#‘𝐴) = 0 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
15	14	a1d 25	. . . . 5 ⊢ ((#‘𝐴) = 0 → (((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
16		orc 399	. . . . . . . . 9 ⊢ ((#‘𝐴) = 1 → ((#‘𝐴) = 1 ∨ 𝐴 = ∅))
17	16	orcd 406	. . . . . . . 8 ⊢ ((#‘𝐴) = 1 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
18	17	a1d 25	. . . . . . 7 ⊢ ((#‘𝐴) = 1 → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
19	18	a1d 25	. . . . . 6 ⊢ ((#‘𝐴) = 1 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
20	19	a1d 25	. . . . 5 ⊢ ((#‘𝐴) = 1 → (((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
21		hasheq0 13015	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ V → ((#‘𝐵) = 0 ↔ 𝐵 = ∅))
22	21	biimpd 218	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ V → ((#‘𝐵) = 0 → 𝐵 = ∅))
23	22	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((#‘𝐵) = 0 → 𝐵 = ∅))
24	23	impcom 445	. . . . . . . . . . . 12 ⊢ (((#‘𝐵) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → 𝐵 = ∅)
25	24	olcd 407	. . . . . . . . . . 11 ⊢ (((#‘𝐵) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ((#‘𝐵) = 1 ∨ 𝐵 = ∅))
26	25	olcd 407	. . . . . . . . . 10 ⊢ (((#‘𝐵) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
27	26	a1d 25	. . . . . . . . 9 ⊢ (((#‘𝐵) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
28	27	ex 449	. . . . . . . 8 ⊢ ((#‘𝐵) = 0 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
29	28	a1d 25	. . . . . . 7 ⊢ ((#‘𝐵) = 0 → (1 < (#‘𝐴) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
30		orc 399	. . . . . . . . . . 11 ⊢ ((#‘𝐵) = 1 → ((#‘𝐵) = 1 ∨ 𝐵 = ∅))
31	30	olcd 407	. . . . . . . . . 10 ⊢ ((#‘𝐵) = 1 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
32	31	a1d 25	. . . . . . . . 9 ⊢ ((#‘𝐵) = 1 → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
33	32	a1d 25	. . . . . . . 8 ⊢ ((#‘𝐵) = 1 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
34	33	a1d 25	. . . . . . 7 ⊢ ((#‘𝐵) = 1 → (1 < (#‘𝐴) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
35	1, 2	frgrawopreglem5 26575	. . . . . . . . . . . 12 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
36	35	3expb 1258	. . . . . . . . . . 11 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (1 < (#‘𝐴) ∧ 1 < (#‘𝐵))) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
37		frisusgra 26519	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 FriendGrph 𝐸 → 𝑉 USGrph 𝐸)
38		simplll 794	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑉 USGrph 𝐸)
39		elrabi 3328	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 ∈ {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} → 𝑎 ∈ 𝑉)
40	39, 1	eleq2s 2706	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝑉)
41	40	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → 𝑎 ∈ 𝑉)
42	41	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑎 ∈ 𝑉)
43	1	rabeq2i 3170	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾))
44	43	simplbi 475	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑉)
45	44	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝑉)
46	45	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑉)
47		simpr1r 1112	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) → 𝑎 ≠ 𝑥)
48	47	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → 𝑎 ≠ 𝑥)
49	48	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑎 ≠ 𝑥)
50	42, 46, 49	3jca 1235	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑎 ≠ 𝑥))
51	2	eleq2i 2680	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ (𝑉 ∖ 𝐴))
52		eldif 3550	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 ∈ (𝑉 ∖ 𝐴) ↔ (𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴))
53	51, 52	bitri 263	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 ∈ 𝑉 ∧ ¬ 𝑏 ∈ 𝐴))
54	53	simplbi 475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 ∈ 𝐵 → 𝑏 ∈ 𝑉)
55	54	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑏 ∈ 𝑉)
56	2	eleq2i 2680	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (𝑉 ∖ 𝐴))
57		eldif 3550	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝑉 ∖ 𝐴) ↔ (𝑦 ∈ 𝑉 ∧ ¬ 𝑦 ∈ 𝐴))
58	56, 57	bitri 263	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ 𝑉 ∧ ¬ 𝑦 ∈ 𝐴))
59	58	simplbi 475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑉)
60	59	ad2antll 761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝑉)
61		simpr1l 1111	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) → 𝑏 ≠ 𝑦)
62	61	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → 𝑏 ≠ 𝑦)
63	62	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑏 ≠ 𝑦)
64	55, 60, 63	3jca 1235	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑏 ≠ 𝑦))
65		prcom 4211	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {𝑥, 𝑏} = {𝑏, 𝑥}
66	65	eleq1i 2679	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({𝑥, 𝑏} ∈ ran 𝐸 ↔ {𝑏, 𝑥} ∈ ran 𝐸)
67	66	biimpi 205	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({𝑥, 𝑏} ∈ ran 𝐸 → {𝑏, 𝑥} ∈ ran 𝐸)
68	67	anim2i 591	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸))
69		prcom 4211	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ {𝑎, 𝑦} = {𝑦, 𝑎}
70	69	eleq1i 2679	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({𝑎, 𝑦} ∈ ran 𝐸 ↔ {𝑦, 𝑎} ∈ ran 𝐸)
71	70	biimpi 205	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({𝑎, 𝑦} ∈ ran 𝐸 → {𝑦, 𝑎} ∈ ran 𝐸)
72	71	anim1i 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸) → ({𝑦, 𝑎} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))
73	72	ancomd 466	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸) → ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸))
74	68, 73	anim12i 588	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸) ∧ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸)))
75	74	3adant1 1072	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸) ∧ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸)))
76	75	ad3antlr 763	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸) ∧ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸)))
77		4cyclusnfrgra 26546	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑎 ≠ 𝑥) ∧ (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑏 ≠ 𝑦)) → ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸) ∧ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸)) → ¬ 𝑉 FriendGrph 𝐸))
78	77	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑎 ≠ 𝑥) ∧ (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑏 ≠ 𝑦)) ∧ (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸) ∧ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸))) → ¬ 𝑉 FriendGrph 𝐸)
79	38, 50, 64, 76, 78	syl31anc 1321	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ¬ 𝑉 FriendGrph 𝐸)
80	79	pm2.21d 117	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑉 USGrph 𝐸 ∧ ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
81	80	exp41 636	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 USGrph 𝐸 → (((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))))
82	81	com25 97	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 USGrph 𝐸 → (𝑉 FriendGrph 𝐸 → ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))))
83	37, 82	mpcom 37	. . . . . . . . . . . . . . 15 ⊢ (𝑉 FriendGrph 𝐸 → ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
84	83	imp 444	. . . . . . . . . . . . . 14 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
85	84	rexlimdvv 3019	. . . . . . . . . . . . 13 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
86	85	rexlimdvva 3020	. . . . . . . . . . . 12 ⊢ (𝑉 FriendGrph 𝐸 → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
87	86	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (1 < (#‘𝐴) ∧ 1 < (#‘𝐵))) → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
88	36, 87	mpd 15	. . . . . . . . . 10 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (1 < (#‘𝐴) ∧ 1 < (#‘𝐵))) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
89	88	expcom 450	. . . . . . . . 9 ⊢ ((1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
90	89	a1d 25	. . . . . . . 8 ⊢ ((1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
91	90	expcom 450	. . . . . . 7 ⊢ (1 < (#‘𝐵) → (1 < (#‘𝐴) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
92	29, 34, 91	3jaoi 1383	. . . . . 6 ⊢ (((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)) → (1 < (#‘𝐴) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
93	92	com12 32	. . . . 5 ⊢ (1 < (#‘𝐴) → (((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
94	15, 20, 93	3jaoi 1383	. . . 4 ⊢ (((#‘𝐴) = 0 ∨ (#‘𝐴) = 1 ∨ 1 < (#‘𝐴)) → (((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
95	94	imp 444	. . 3 ⊢ ((((#‘𝐴) = 0 ∨ (#‘𝐴) = 1 ∨ 1 < (#‘𝐴)) ∧ ((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵))) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
96	6, 95	mpcom 37	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
97	3, 96	mpcom 37	1 ⊢ (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   < clt 9953  #chash 12979   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-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-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-usgra 25862  df-nbgra 25949  df-vdgr 26421  df-frgra 26516

This theorem is referenced by:  frgraregorufr0  26579