usgrcyclnl2 - Metamath Proof Explorer

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Theorem usgrcyclnl2 26169

Description: In an undirected simple graph (with no loops!) there are no cycles with length 2 (consisting of two edges ). (Contributed by Alexander van der Vekens, 9-Nov-2017.)

**Assertion**
Ref	Expression
usgrcyclnl2	⊢ ((𝑉 USGrph 𝐸 ∧ 𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 2)

Proof of Theorem usgrcyclnl2 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cyclispth 26157	. . . 4 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → 𝐹(𝑉 Paths 𝐸)𝑃)
2		pthistrl 26102	. . . . 5 ⊢ (𝐹(𝑉 Paths 𝐸)𝑃 → 𝐹(𝑉 Trails 𝐸)𝑃)
3		cycliswlk 26160	. . . . . 6 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → 𝐹(𝑉 Walks 𝐸)𝑃)
4		wlkbprop 26051	. . . . . 6 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ₀ ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
5		istrl2 26068	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
6		pm3.2 462	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
7	5, 6	sylbid 229	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
8	7	3adant1 1072	. . . . . 6 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
9	3, 4, 8	3syl 18	. . . . 5 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝐹(𝑉 Trails 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
10	2, 9	syl5com 31	. . . 4 ⊢ (𝐹(𝑉 Paths 𝐸)𝑃 → (𝐹(𝑉 Cycles 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
11	1, 10	mpcom 37	. . 3 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
12		iscycl 26153	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ↔ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
13	12	adantr 480	. . . 4 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → (𝐹(𝑉 Cycles 𝐸)𝑃 ↔ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
14		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
15		f1eq2 6010	. . . . . . . . . . . . . 14 ⊢ ((0..^(#‘𝐹)) = (0..^2) → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ 𝐹:(0..^2)–1-1→dom 𝐸))
16	14, 15	syl 17	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 2 → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ 𝐹:(0..^2)–1-1→dom 𝐸))
17	14	raleqdv 3121	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 2 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
18	16, 17	anbi12d 743	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 2 → ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
19		fveq2 6103	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 2 → (𝑃‘(#‘𝐹)) = (𝑃‘2))
20	19	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 2 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃‘2)))
21	18, 20	anbi12d 743	. . . . . . . . . . 11 ⊢ ((#‘𝐹) = 2 → (((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ↔ ((𝐹:(0..^2)–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘2))))
22	21	anbi1d 737	. . . . . . . . . 10 ⊢ ((#‘𝐹) = 2 → ((((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ∧ 𝑉 USGrph 𝐸) ↔ (((𝐹:(0..^2)–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘2)) ∧ 𝑉 USGrph 𝐸)))
23		fzo0to2pr 12420	. . . . . . . . . . . . . 14 ⊢ (0..^2) = {0, 1}
24	23	raleqi 3119	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ (0..^2)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
25		2wlklem 26094	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
26		prcom 4211	. . . . . . . . . . . . . . . . . . 19 ⊢ {(𝑃‘1), (𝑃‘2)} = {(𝑃‘2), (𝑃‘1)}
27	26	eqeq2i 2622	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)})
28		preq1 4212	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘2) = (𝑃‘0) → {(𝑃‘2), (𝑃‘1)} = {(𝑃‘0), (𝑃‘1)})
29	28	eqcoms 2618	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘2), (𝑃‘1)} = {(𝑃‘0), (𝑃‘1)})
30	29	eqeq2d 2620	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃‘0) = (𝑃‘2) → ((𝐸‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
31	27, 30	syl5bb 271	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃‘0) = (𝑃‘2) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
32	31	anbi2d 736	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃‘0) = (𝑃‘2) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)})))
33		eqtr3 2631	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}) → (𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)))
34		usgraf1 25889	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→ran 𝐸)
35		f1f 6014	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:(0..^2)–1-1→dom 𝐸 → 𝐹:(0..^2)⟶dom 𝐸)
36		2nn 11062	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2 ∈ ℕ
37		lbfzo0 12375	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
38	36, 37	mpbir 220	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ (0..^2)
39		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:(0..^2)⟶dom 𝐸 ∧ 0 ∈ (0..^2)) → (𝐹‘0) ∈ dom 𝐸)
40	38, 39	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:(0..^2)⟶dom 𝐸 → (𝐹‘0) ∈ dom 𝐸)
41		1nn0 11185	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ∈ ℕ₀
42		1lt2 11071	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 < 2
43		elfzo0 12376	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ ∧ 1 < 2))
44	41, 36, 42, 43	mpbir3an 1237	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ (0..^2)
45		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:(0..^2)⟶dom 𝐸 ∧ 1 ∈ (0..^2)) → (𝐹‘1) ∈ dom 𝐸)
46	44, 45	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:(0..^2)⟶dom 𝐸 → (𝐹‘1) ∈ dom 𝐸)
47	40, 46	jca 553	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:(0..^2)⟶dom 𝐸 → ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸))
48	35, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸))
49		f1fveq 6420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸)) → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) ↔ (𝐹‘0) = (𝐹‘1)))
50	48, 49	sylan2 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐹:(0..^2)–1-1→dom 𝐸) → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) ↔ (𝐹‘0) = (𝐹‘1)))
51		f1fveq 6420	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:(0..^2)–1-1→dom 𝐸 ∧ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))) → ((𝐹‘0) = (𝐹‘1) ↔ 0 = 1))
52	38, 44, 51	mpanr12 717	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝐹‘0) = (𝐹‘1) ↔ 0 = 1))
53		ax-1ne0 9884	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ≠ 0
54		necom 2835	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ≠ 0 ↔ 0 ≠ 1)
55		df-ne 2782	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0 ≠ 1 ↔ ¬ 0 = 1)
56		pm2.21 119	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (¬ 0 = 1 → (0 = 1 → (#‘𝐹) ≠ 2))
57	55, 56	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 ≠ 1 → (0 = 1 → (#‘𝐹) ≠ 2))
58	54, 57	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ≠ 0 → (0 = 1 → (#‘𝐹) ≠ 2))
59	53, 58	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 = 1 → (#‘𝐹) ≠ 2)
60	52, 59	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝐹‘0) = (𝐹‘1) → (#‘𝐹) ≠ 2))
61	60	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐹:(0..^2)–1-1→dom 𝐸) → ((𝐹‘0) = (𝐹‘1) → (#‘𝐹) ≠ 2))
62	50, 61	sylbid 229	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐹:(0..^2)–1-1→dom 𝐸) → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → (#‘𝐹) ≠ 2))
63	62	ex 449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → (#‘𝐹) ≠ 2)))
64	34, 63	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 USGrph 𝐸 → (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → (#‘𝐹) ≠ 2)))
65	64	com13 86	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → (𝐹:(0..^2)–1-1→dom 𝐸 → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
66	33, 65	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}) → (𝐹:(0..^2)–1-1→dom 𝐸 → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
67	32, 66	syl6bi 242	. . . . . . . . . . . . . . 15 ⊢ ((𝑃‘0) = (𝑃‘2) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐹:(0..^2)–1-1→dom 𝐸 → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2))))
68	67	com3l 87	. . . . . . . . . . . . . 14 ⊢ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝑃‘0) = (𝑃‘2) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2))))
69	25, 68	sylbi 206	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝑃‘0) = (𝑃‘2) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2))))
70	24, 69	sylbi 206	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (0..^2)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝑃‘0) = (𝑃‘2) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2))))
71	70	impcom 445	. . . . . . . . . . 11 ⊢ ((𝐹:(0..^2)–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘2) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
72	71	imp31 447	. . . . . . . . . 10 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘2)) ∧ 𝑉 USGrph 𝐸) → (#‘𝐹) ≠ 2)
73	22, 72	syl6bi 242	. . . . . . . . 9 ⊢ ((#‘𝐹) = 2 → ((((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ∧ 𝑉 USGrph 𝐸) → (#‘𝐹) ≠ 2))
74		ax-1 6	. . . . . . . . 9 ⊢ ((#‘𝐹) ≠ 2 → ((((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ∧ 𝑉 USGrph 𝐸) → (#‘𝐹) ≠ 2))
75	73, 74	pm2.61ine 2865	. . . . . . . 8 ⊢ ((((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ∧ 𝑉 USGrph 𝐸) → (#‘𝐹) ≠ 2)
76	75	exp31 628	. . . . . . 7 ⊢ ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
77	76	3adant2 1073	. . . . . 6 ⊢ ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
78	77	adantld 482	. . . . 5 ⊢ ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
79	78	adantl 481	. . . 4 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
80	13, 79	sylbid 229	. . 3 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
81	11, 80	mpcom 37	. 2 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2))
82	81	impcom 445	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 2)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 {cpr 4127 class class class wbr 4583 dom cdm 5038 ran crn 5039 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 USGrph cusg 25859 Walks cwalk 26026 Trails ctrail 26027 Paths cpath 26028 Cycles ccycl 26035

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-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-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 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-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-usgra 25862 df-wlk 26036 df-trail 26037 df-pth 26038 df-cycl 26041

This theorem is referenced by: (None)

W3C validator