Theorem usgr2trlncl 27847

Description: In a simple graph, any trail of length 2 does not start and end at the same vertex. (Contributed by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.)

Assertion

Ref	Expression
usgr2trlncl	⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2)))

Proof of Theorem usgr2trlncl

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrupgr 27273	. . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
2		eqid 2737	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2737	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	upgrf1istrl 27791	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
5	1, 4	syl 17	. . . 4 ⊢ (𝐺 ∈ USGraph → (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
6		eqidd 2738	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → 𝐹 = 𝐹)
7		oveq2 7221	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
8		fzo0to2pr 13327	. . . . . . . . . . . . 13 ⊢ (0..^2) = {0, 1}
9	7, 8	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
10		eqidd 2738	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
11	6, 9, 10	f1eq123d 6653	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ 𝐹:{0, 1}–1-1→dom (iEdg‘𝐺)))
12	9	raleqdv 3325	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → (∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
13		2wlklem 27755	. . . . . . . . . . . 12 ⊢ (∀𝑖 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
14	12, 13	bitrdi 290	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 2 → (∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
15	11, 14	anbi12d 634	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 2 → ((𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ↔ (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
16	15	adantl 485	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ↔ (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
17		c0ex 10827	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
18		1ex 10829	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
19	17, 18	pm3.2i 474	. . . . . . . . . . . . 13 ⊢ (0 ∈ V ∧ 1 ∈ V)
20		0ne1 11901	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
21		eqid 2737	. . . . . . . . . . . . . 14 ⊢ {0, 1} = {0, 1}
22	21	f12dfv 7084	. . . . . . . . . . . . 13 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ 0 ≠ 1) → (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ↔ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1))))
23	19, 20, 22	mp2an 692	. . . . . . . . . . . 12 ⊢ (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ↔ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)))
24		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
25	3, 24	usgrf1oedg 27295	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
26		f1of1 6660	. . . . . . . . . . . . . 14 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
27		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 𝐹:{0, 1}⟶dom (iEdg‘𝐺))
28	17	prid1 4678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ {0, 1}
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 0 ∈ {0, 1})
30	27, 29	ffvelrnd 6905	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → (𝐹‘0) ∈ dom (iEdg‘𝐺))
31	18	prid2 4679	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ {0, 1}
32	31	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 1 ∈ {0, 1})
33	27, 32	ffvelrnd 6905	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → (𝐹‘1) ∈ dom (iEdg‘𝐺))
34	30, 33	jca 515	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → ((𝐹‘0) ∈ dom (iEdg‘𝐺) ∧ (𝐹‘1) ∈ dom (iEdg‘𝐺)))
35	34	anim1ci 619	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)) → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) ∧ ((𝐹‘0) ∈ dom (iEdg‘𝐺) ∧ (𝐹‘1) ∈ dom (iEdg‘𝐺))))
36		f1veqaeq 7069	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) ∧ ((𝐹‘0) ∈ dom (iEdg‘𝐺) ∧ (𝐹‘1) ∈ dom (iEdg‘𝐺))) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → (𝐹‘0) = (𝐹‘1)))
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → (𝐹‘0) = (𝐹‘1)))
38	37	necon3d 2961	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)) → ((𝐹‘0) ≠ (𝐹‘1) → ((iEdg‘𝐺)‘(𝐹‘0)) ≠ ((iEdg‘𝐺)‘(𝐹‘1))))
39		simpl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)})
40		simpr 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})
41	39, 40	neeq12d 3002	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (((iEdg‘𝐺)‘(𝐹‘0)) ≠ ((iEdg‘𝐺)‘(𝐹‘1)) ↔ {(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)}))
42		preq1 4649	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘0), (𝑃‘1)} = {(𝑃‘2), (𝑃‘1)})
43		prcom 4648	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {(𝑃‘2), (𝑃‘1)} = {(𝑃‘1), (𝑃‘2)}
44	42, 43	eqtrdi 2794	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘0), (𝑃‘1)} = {(𝑃‘1), (𝑃‘2)})
45	44	necon3i 2973	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)} → (𝑃‘0) ≠ (𝑃‘2))
46	41, 45	syl6bi 256	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (((iEdg‘𝐺)‘(𝐹‘0)) ≠ ((iEdg‘𝐺)‘(𝐹‘1)) → (𝑃‘0) ≠ (𝑃‘2)))
47	46	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘(𝐹‘0)) ≠ ((iEdg‘𝐺)‘(𝐹‘1)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))
48	47	a1d 25	. . . . . . . . . . . . . . . . . 18 ⊢ (((iEdg‘𝐺)‘(𝐹‘0)) ≠ ((iEdg‘𝐺)‘(𝐹‘1)) → (𝐺 ∈ USGraph → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2))))
49	38, 48	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)) → ((𝐹‘0) ≠ (𝐹‘1) → (𝐺 ∈ USGraph → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))))
50	49	expcom 417	. . . . . . . . . . . . . . . 16 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) → (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → ((𝐹‘0) ≠ (𝐹‘1) → (𝐺 ∈ USGraph → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2))))))
51	50	impd 414	. . . . . . . . . . . . . . 15 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) → ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)) → (𝐺 ∈ USGraph → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))))
52	51	com23 86	. . . . . . . . . . . . . 14 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) → (𝐺 ∈ USGraph → ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))))
53	26, 52	syl 17	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (𝐺 ∈ USGraph → ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))))
54	25, 53	mpcom 38	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2))))
55	23, 54	syl5bi 245	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2))))
56	55	impd 414	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → ((𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2)))
57	56	adantr 484	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ((𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2)))
58	16, 57	sylbid 243	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃‘0) ≠ (𝑃‘2)))
59	58	com12 32	. . . . . . 7 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝑃‘0) ≠ (𝑃‘2)))
60	59	3adant2 1133	. . . . . 6 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝑃‘0) ≠ (𝑃‘2)))
61	60	expdcom 418	. . . . 5 ⊢ (𝐺 ∈ USGraph → ((♯‘𝐹) = 2 → ((𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃‘0) ≠ (𝑃‘2))))
62	61	com23 86	. . . 4 ⊢ (𝐺 ∈ USGraph → ((𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘2))))
63	5, 62	sylbid 243	. . 3 ⊢ (𝐺 ∈ USGraph → (𝐹(Trails‘𝐺)𝑃 → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘2))))
64	63	com23 86	. 2 ⊢ (𝐺 ∈ USGraph → ((♯‘𝐹) = 2 → (𝐹(Trails‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2))))
65	64	imp 410	1 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  Vcvv 3408  {cpr 4543   class class class wbr 5053  dom cdm 5551  ⟶wf 6376  –1-1→wf1 6377  –1-1-onto→wf1o 6379  ‘cfv 6380  (class class class)co 7213  0cc0 10729  1c1 10730   + caddc 10732  2c2 11885  ...cfz 13095  ..^cfzo 13238  ♯chash 13896  Vtxcvtx 27087  iEdgciedg 27088  Edgcedg 27138  UPGraphcupgr 27171  USGraphcusgr 27240  Trailsctrls 27778

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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ifp 1064  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-oadd 8206  df-er 8391  df-map 8510  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-fz 13096  df-fzo 13239  df-hash 13897  df-word 14070  df-edg 27139  df-uhgr 27149  df-upgr 27173  df-uspgr 27241  df-usgr 27242  df-wlks 27687  df-trls 27780

This theorem is referenced by:  usgr2trlspth  27848  usgr2trlncrct  27890