upgr3v3e3cycl - Metamath Proof Explorer

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Theorem upgr3v3e3cycl 30010

Description: If there is a cycle of length 3 in a pseudograph, there are three distinct vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)

**Hypotheses**
Ref	Expression
upgr3v3e3cycl.e	⊢ 𝐸 = (Edg‘𝐺)
upgr3v3e3cycl.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
upgr3v3e3cycl	⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ (♯‘𝐹) = 3) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))

Distinct variable groups: 𝐸,𝑎,𝑏,𝑐 𝑃,𝑎,𝑏,𝑐 𝑉,𝑎,𝑏,𝑐 Allowed substitution hints: 𝐹(𝑎,𝑏,𝑐) 𝐺(𝑎,𝑏,𝑐)

Proof of Theorem upgr3v3e3cycl Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cyclprop 29627	. . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2		pthiswlk 29561	. . . . 5 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		upgr3v3e3cycl.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
4	3	upgrwlkvtxedg 29479	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸)
5		fveq2 6902	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 3 → (𝑃‘(♯‘𝐹)) = (𝑃‘3))
6	5	eqeq2d 2739	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) = (𝑃‘3)))
7	6	anbi2d 628	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 3 → ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3))))
8		oveq2 7434	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 3 → (0..^(♯‘𝐹)) = (0..^3))
9		fzo0to3tp 13758	. . . . . . . . . . . . . . . 16 ⊢ (0..^3) = {0, 1, 2}
10	8, 9	eqtrdi 2784	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 3 → (0..^(♯‘𝐹)) = {0, 1, 2})
11	10	raleqdv 3323	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 3 → (∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ ∀𝑘 ∈ {0, 1, 2} {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸))
12		c0ex 11246	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
13		1ex 11248	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
14		2ex 12327	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ V
15		fveq2 6902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
16		fv0p1e1 12373	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1))
17	15, 16	preq12d 4750	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘0), (𝑃‘1)})
18	17	eleq1d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ {(𝑃‘0), (𝑃‘1)} ∈ 𝐸))
19		fveq2 6902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1))
20		oveq1 7433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 1 → (𝑘 + 1) = (1 + 1))
21		1p1e2 12375	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 + 1) = 2
22	20, 21	eqtrdi 2784	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 1 → (𝑘 + 1) = 2)
23	22	fveq2d 6906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2))
24	19, 23	preq12d 4750	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 1 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘1), (𝑃‘2)})
25	24	eleq1d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸))
26		fveq2 6902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
27		oveq1 7433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 2 → (𝑘 + 1) = (2 + 1))
28		2p1e3 12392	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 1) = 3
29	27, 28	eqtrdi 2784	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 2 → (𝑘 + 1) = 3)
30	29	fveq2d 6906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 2 → (𝑃‘(𝑘 + 1)) = (𝑃‘3))
31	26, 30	preq12d 4750	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 2 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘2), (𝑃‘3)})
32	31	eleq1d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 2 → ({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))
33	12, 13, 14, 18, 25, 32	raltp 4714	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ {0, 1, 2} {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))
34	11, 33	bitrdi 286	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 3 → (∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸)))
35	7, 34	anbi12d 630	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 3 → (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸) ↔ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))))
36		upgr3v3e3cycl.v	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑉 = (Vtx‘𝐺)
37	36	wlkp 29450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
38		oveq2 7434	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) = 3 → (0...(♯‘𝐹)) = (0...3))
39	38	feq2d 6713	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) = 3 → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...3)⟶𝑉))
40		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃:(0...3)⟶𝑉 → 𝑃:(0...3)⟶𝑉)
41		3nn0 12528	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 3 ∈ ℕ₀
42		0elfz 13638	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (3 ∈ ℕ₀ → 0 ∈ (0...3))
43	41, 42	mp1i 13	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃:(0...3)⟶𝑉 → 0 ∈ (0...3))
44	40, 43	ffvelcdmd 7100	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃:(0...3)⟶𝑉 → (𝑃‘0) ∈ 𝑉)
45		1nn0 12526	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℕ₀
46		1lt3 12423	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 < 3
47		fvffz0 13659	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((3 ∈ ℕ₀ ∧ 1 ∈ ℕ₀ ∧ 1 < 3) ∧ 𝑃:(0...3)⟶𝑉) → (𝑃‘1) ∈ 𝑉)
48	47	ex 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((3 ∈ ℕ₀ ∧ 1 ∈ ℕ₀ ∧ 1 < 3) → (𝑃:(0...3)⟶𝑉 → (𝑃‘1) ∈ 𝑉))
49	41, 45, 46, 48	mp3an 1457	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃:(0...3)⟶𝑉 → (𝑃‘1) ∈ 𝑉)
50		2nn0 12527	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℕ₀
51		2lt3 12422	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 < 3
52		fvffz0 13659	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((3 ∈ ℕ₀ ∧ 2 ∈ ℕ₀ ∧ 2 < 3) ∧ 𝑃:(0...3)⟶𝑉) → (𝑃‘2) ∈ 𝑉)
53	52	ex 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((3 ∈ ℕ₀ ∧ 2 ∈ ℕ₀ ∧ 2 < 3) → (𝑃:(0...3)⟶𝑉 → (𝑃‘2) ∈ 𝑉))
54	41, 50, 51, 53	mp3an 1457	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃:(0...3)⟶𝑉 → (𝑃‘2) ∈ 𝑉)
55	44, 49, 54	3jca 1125	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃:(0...3)⟶𝑉 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))
56	39, 55	biimtrdi 252	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐹) = 3 → (𝑃:(0...(♯‘𝐹))⟶𝑉 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
57	56	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 → ((♯‘𝐹) = 3 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
58	2, 37, 57	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((♯‘𝐹) = 3 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
59	58	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → ((♯‘𝐹) = 3 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
60	59	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸)) → ((♯‘𝐹) = 3 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
61	60	impcom 406	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) = 3 ∧ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))) → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))
62		preq2 4743	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘3) = (𝑃‘0) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘2), (𝑃‘0)})
63	62	eqcoms 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘2), (𝑃‘0)})
64	63	adantl 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘2), (𝑃‘0)})
65	64	eleq1d 2814	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → ({(𝑃‘2), (𝑃‘3)} ∈ 𝐸 ↔ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸))
66	65	3anbi3d 1438	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → (({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸) ↔ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸)))
67	66	biimpa 475	. . . . . . . . . . . . . . 15 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸)) → ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸))
68	67	adantl 480	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) = 3 ∧ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))) → ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸))
69		simpll 765	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → 𝐹(Paths‘𝐺)𝑃)
70		breq2 5156	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) = 3 → (1 < (♯‘𝐹) ↔ 1 < 3))
71	46, 70	mpbiri 257	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) = 3 → 1 < (♯‘𝐹))
72	71	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → 1 < (♯‘𝐹))
73		3nn 12329	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 3 ∈ ℕ
74		lbfzo0 13712	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 ∈ (0..^3) ↔ 3 ∈ ℕ)
75	73, 74	mpbir 230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ (0..^3)
76	75, 8	eleqtrrid 2836	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) = 3 → 0 ∈ (0..^(♯‘𝐹)))
77	76	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → 0 ∈ (0..^(♯‘𝐹)))
78		pthdadjvtx 29564	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹) ∧ 0 ∈ (0..^(♯‘𝐹))) → (𝑃‘0) ≠ (𝑃‘(0 + 1)))
79		1e0p1 12757	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 = (0 + 1)
80	79	fveq2i 6905	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃‘1) = (𝑃‘(0 + 1))
81	80	neeq2i 3003	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘0) ≠ (𝑃‘1) ↔ (𝑃‘0) ≠ (𝑃‘(0 + 1)))
82	78, 81	sylibr 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹) ∧ 0 ∈ (0..^(♯‘𝐹))) → (𝑃‘0) ≠ (𝑃‘1))
83	69, 72, 77, 82	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → (𝑃‘0) ≠ (𝑃‘1))
84		elfzo0 13713	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 ∈ (0..^3) ↔ (1 ∈ ℕ₀ ∧ 3 ∈ ℕ ∧ 1 < 3))
85	45, 73, 46, 84	mpbir3an 1338	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ (0..^3)
86	85, 8	eleqtrrid 2836	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) = 3 → 1 ∈ (0..^(♯‘𝐹)))
87	86	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → 1 ∈ (0..^(♯‘𝐹)))
88		pthdadjvtx 29564	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹) ∧ 1 ∈ (0..^(♯‘𝐹))) → (𝑃‘1) ≠ (𝑃‘(1 + 1)))
89		df-2 12313	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 = (1 + 1)
90	89	fveq2i 6905	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃‘2) = (𝑃‘(1 + 1))
91	90	neeq2i 3003	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘1) ≠ (𝑃‘2) ↔ (𝑃‘1) ≠ (𝑃‘(1 + 1)))
92	88, 91	sylibr 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹) ∧ 1 ∈ (0..^(♯‘𝐹))) → (𝑃‘1) ≠ (𝑃‘2))
93	69, 72, 87, 92	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → (𝑃‘1) ≠ (𝑃‘2))
94		elfzo0 13713	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 ∈ (0..^3) ↔ (2 ∈ ℕ₀ ∧ 3 ∈ ℕ ∧ 2 < 3))
95	50, 73, 51, 94	mpbir3an 1338	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ (0..^3)
96	95, 8	eleqtrrid 2836	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) = 3 → 2 ∈ (0..^(♯‘𝐹)))
97	96	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → 2 ∈ (0..^(♯‘𝐹)))
98		pthdadjvtx 29564	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹) ∧ 2 ∈ (0..^(♯‘𝐹))) → (𝑃‘2) ≠ (𝑃‘(2 + 1)))
99	69, 72, 97, 98	syl3anc 1368	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → (𝑃‘2) ≠ (𝑃‘(2 + 1)))
100		neeq2 3001	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘3) → ((𝑃‘2) ≠ (𝑃‘0) ↔ (𝑃‘2) ≠ (𝑃‘3)))
101		df-3 12314	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 3 = (2 + 1)
102	101	fveq2i 6905	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃‘3) = (𝑃‘(2 + 1))
103	102	neeq2i 3003	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘2) ≠ (𝑃‘3) ↔ (𝑃‘2) ≠ (𝑃‘(2 + 1)))
104	100, 103	bitrdi 286	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘0) = (𝑃‘3) → ((𝑃‘2) ≠ (𝑃‘0) ↔ (𝑃‘2) ≠ (𝑃‘(2 + 1))))
105	104	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → ((𝑃‘2) ≠ (𝑃‘0) ↔ (𝑃‘2) ≠ (𝑃‘(2 + 1))))
106	105	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → ((𝑃‘2) ≠ (𝑃‘0) ↔ (𝑃‘2) ≠ (𝑃‘(2 + 1))))
107	99, 106	mpbird 256	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → (𝑃‘2) ≠ (𝑃‘0))
108	83, 93, 107	3jca 1125	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (♯‘𝐹) = 3) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0)))
109	108	ex 411	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → ((♯‘𝐹) = 3 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0))))
110	109	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸)) → ((♯‘𝐹) = 3 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0))))
111	110	impcom 406	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) = 3 ∧ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0)))
112		preq1 4742	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝑃‘0) → {𝑎, 𝑏} = {(𝑃‘0), 𝑏})
113	112	eleq1d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑃‘0) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑃‘0), 𝑏} ∈ 𝐸))
114		preq2 4743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝑃‘0) → {𝑐, 𝑎} = {𝑐, (𝑃‘0)})
115	114	eleq1d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑃‘0) → ({𝑐, 𝑎} ∈ 𝐸 ↔ {𝑐, (𝑃‘0)} ∈ 𝐸))
116	113, 115	3anbi13d 1434	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝑃‘0) → (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ↔ ({(𝑃‘0), 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸)))
117		neeq1 3000	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑃‘0) → (𝑎 ≠ 𝑏 ↔ (𝑃‘0) ≠ 𝑏))
118		neeq2 3001	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑃‘0) → (𝑐 ≠ 𝑎 ↔ 𝑐 ≠ (𝑃‘0)))
119	117, 118	3anbi13d 1434	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝑃‘0) → ((𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎) ↔ ((𝑃‘0) ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0))))
120	116, 119	anbi12d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝑃‘0) → ((({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)) ↔ (({(𝑃‘0), 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0)))))
121		preq2 4743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑃‘1) → {(𝑃‘0), 𝑏} = {(𝑃‘0), (𝑃‘1)})
122	121	eleq1d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑃‘1) → ({(𝑃‘0), 𝑏} ∈ 𝐸 ↔ {(𝑃‘0), (𝑃‘1)} ∈ 𝐸))
123		preq1 4742	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑃‘1) → {𝑏, 𝑐} = {(𝑃‘1), 𝑐})
124	123	eleq1d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑃‘1) → ({𝑏, 𝑐} ∈ 𝐸 ↔ {(𝑃‘1), 𝑐} ∈ 𝐸))
125	122, 124	3anbi12d 1433	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑃‘1) → (({(𝑃‘0), 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ↔ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸)))
126		neeq2 3001	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑃‘1) → ((𝑃‘0) ≠ 𝑏 ↔ (𝑃‘0) ≠ (𝑃‘1)))
127		neeq1 3000	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑃‘1) → (𝑏 ≠ 𝑐 ↔ (𝑃‘1) ≠ 𝑐))
128	126, 127	3anbi12d 1433	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑃‘1) → (((𝑃‘0) ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0)) ↔ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0))))
129	125, 128	anbi12d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑃‘1) → ((({(𝑃‘0), 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0))) ↔ (({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0)))))
130		preq2 4743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝑃‘2) → {(𝑃‘1), 𝑐} = {(𝑃‘1), (𝑃‘2)})
131	130	eleq1d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑃‘2) → ({(𝑃‘1), 𝑐} ∈ 𝐸 ↔ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸))
132		preq1 4742	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝑃‘2) → {𝑐, (𝑃‘0)} = {(𝑃‘2), (𝑃‘0)})
133	132	eleq1d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑃‘2) → ({𝑐, (𝑃‘0)} ∈ 𝐸 ↔ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸))
134	131, 133	3anbi23d 1435	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑃‘2) → (({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ↔ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸)))
135		neeq2 3001	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑃‘2) → ((𝑃‘1) ≠ 𝑐 ↔ (𝑃‘1) ≠ (𝑃‘2)))
136		neeq1 3000	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑃‘2) → (𝑐 ≠ (𝑃‘0) ↔ (𝑃‘2) ≠ (𝑃‘0)))
137	135, 136	3anbi23d 1435	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑃‘2) → (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0)) ↔ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0))))
138	134, 137	anbi12d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑃‘2) → ((({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0))) ↔ (({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0)))))
139	120, 129, 138	rspc3ev 3628	. . . . . . . . . . . . . 14 ⊢ ((((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉) ∧ (({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0)))) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))
140	61, 68, 111, 139	syl12anc 835	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) = 3 ∧ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))
141	140	ex 411	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 3 → (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))))
142	35, 141	sylbid 239	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 3 → (((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))))
143	142	expd 414	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 3 → ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))
144	143	com13 88	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 → ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((♯‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))
145	4, 144	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((♯‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))
146	145	expcom 412	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((♯‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))))))
147	146	com23 86	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝐺 ∈ UPGraph → ((♯‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))))))
148	147	expd 414	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹(Paths‘𝐺)𝑃 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (𝐺 ∈ UPGraph → ((♯‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))))
149	2, 148	mpcom 38	. . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (𝐺 ∈ UPGraph → ((♯‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))))))
150	149	imp 405	. . 3 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝐺 ∈ UPGraph → ((♯‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))
151	1, 150	syl 17	. 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (𝐺 ∈ UPGraph → ((♯‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))
152	151	3imp21 1111	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ (♯‘𝐹) = 3) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {cpr 4634 {ctp 4636 class class class wbr 5152 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 0cc0 11146 1c1 11147 + caddc 11149 < clt 11286 ℕcn 12250 2c2 12305 3c3 12306 ℕ₀cn0 12510 ...cfz 13524 ..^cfzo 13667 ♯chash 14329 Vtxcvtx 28829 Edgcedg 28880 UPGraphcupgr 28913 Walkscwlks 29430 Pathscpths 29546 Cyclesccycls 29619

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-xnn0 12583 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-edg 28881 df-uhgr 28891 df-upgr 28915 df-wlks 29433 df-trls 29526 df-pths 29550 df-cycls 29621

This theorem is referenced by: umgr3v3e3cycl 30014

W3C validator