Theorem usgr2trlncl 29697

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 29119	. . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
2		eqid 2730	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2730	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	upgrf1istrl 29638	. . . . 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 2731	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → 𝐹 = 𝐹)
7		oveq2 7398	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
8		fzo0to2pr 13718	. . . . . . . . . . . . 13 ⊢ (0..^2) = {0, 1}
9	7, 8	eqtrdi 2781	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
10		eqidd 2731	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
11	6, 9, 10	f1eq123d 6795	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ 𝐹:{0, 1}–1-1→dom (iEdg‘𝐺)))
12	9	raleqdv 3301	. . . . . . . . . . . 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 29602	. . . . . . . . . . . 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 287	. . . . . . . . . . 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 632	. . . . . . . . . 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 481	. . . . . . . . 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 11175	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
18		1ex 11177	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
19	17, 18	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ (0 ∈ V ∧ 1 ∈ V)
20		0ne1 12264	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
21		eqid 2730	. . . . . . . . . . . . . 14 ⊢ {0, 1} = {0, 1}
22	21	f12dfv 7251	. . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
25	3, 24	usgrf1oedg 29141	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
26		f1of1 6802	. . . . . . . . . . . . . 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 4729	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ {0, 1}
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 0 ∈ {0, 1})
30	27, 29	ffvelcdmd 7060	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → (𝐹‘0) ∈ dom (iEdg‘𝐺))
31	18	prid2 4730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ {0, 1}
32	31	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 1 ∈ {0, 1})
33	27, 32	ffvelcdmd 7060	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → (𝐹‘1) ∈ dom (iEdg‘𝐺))
34	30, 33	jca 511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → ((𝐹‘0) ∈ dom (iEdg‘𝐺) ∧ (𝐹‘1) ∈ dom (iEdg‘𝐺)))
35	34	anim1ci 616	. . . . . . . . . . . . . . . . . . . 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 7234	. . . . . . . . . . . . . . . . . . . 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 2947	. . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . 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 2987	. . . . . . . . . . . . . . . . . . . . 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 4700	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘0), (𝑃‘1)} = {(𝑃‘2), (𝑃‘1)})
43		prcom 4699	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {(𝑃‘2), (𝑃‘1)} = {(𝑃‘1), (𝑃‘2)}
44	42, 43	eqtrdi 2781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘0), (𝑃‘1)} = {(𝑃‘1), (𝑃‘2)})
45	44	necon3i 2958	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)} → (𝑃‘0) ≠ (𝑃‘2))
46	41, 45	biimtrdi 253	. . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . 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 410	. . . . . . . . . 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 480	. . . . . . . . 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 240	. . . . . . . 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 1131	. . . . . 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 414	. . . . 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 240	. . 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 406	1 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  Vcvv 3450  {cpr 4594   class class class wbr 5110  dom cdm 5641  ⟶wf 6510  –1-1→wf1 6511  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  0cc0 11075  1c1 11076   + caddc 11078  2c2 12248  ...cfz 13475  ..^cfzo 13622  ♯chash 14302  Vtxcvtx 28930  iEdgciedg 28931  Edgcedg 28981  UPGraphcupgr 29014  USGraphcusgr 29083  Trailsctrls 29625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-fz 13476  df-fzo 13623  df-hash 14303  df-word 14486  df-edg 28982  df-uhgr 28992  df-upgr 29016  df-uspgr 29084  df-usgr 29085  df-wlks 29534  df-trls 29627

This theorem is referenced by:  usgr2trlspth  29698  usgr2trlncrct  29743