Theorem usgr2trlncl 29899

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 29325	. . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
2		eqid 2756	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2756	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	upgrf1istrl 29841	. . . . 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 2757	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → 𝐹 = 𝐹)
7		oveq2 7393	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
8		fzo0to2pr 13746	. . . . . . . . . . . . 13 ⊢ (0..^2) = {0, 1}
9	7, 8	eqtrdi 2807	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
10		eqidd 2757	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
11	6, 9, 10	f1eq123d 6787	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ 𝐹:{0, 1}–1-1→dom (iEdg‘𝐺)))
12	9	raleqdv 3314	. . . . . . . . . . . 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 29805	. . . . . . . . . . . 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 289	. . . . . . . . . . 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 640	. . . . . . . . . 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 484	. . . . . . . . 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 11163	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
18		1ex 11166	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
19	17, 18	pm3.2i 473	. . . . . . . . . . . . 13 ⊢ (0 ∈ V ∧ 1 ∈ V)
20		0ne1 12279	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
21		eqid 2756	. . . . . . . . . . . . . 14 ⊢ {0, 1} = {0, 1}
22	21	f12dfv 7246	. . . . . . . . . . . . 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 700	. . . . . . . . . . . 12 ⊢ (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ↔ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)))
24		eqid 2756	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
25	3, 24	usgrf1oedg 29347	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
26		f1of1 6794	. . . . . . . . . . . . . 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 4715	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ {0, 1}
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 0 ∈ {0, 1})
30	27, 29	ffvelcdmd 7055	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → (𝐹‘0) ∈ dom (iEdg‘𝐺))
31	18	prid2 4716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ {0, 1}
32	31	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 1 ∈ {0, 1})
33	27, 32	ffvelcdmd 7055	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → (𝐹‘1) ∈ dom (iEdg‘𝐺))
34	30, 33	jca 518	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → ((𝐹‘0) ∈ dom (iEdg‘𝐺) ∧ (𝐹‘1) ∈ dom (iEdg‘𝐺)))
35	34	anim1ci 624	. . . . . . . . . . . . . . . . . . . 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 7229	. . . . . . . . . . . . . . . . . . . 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 2972	. . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . 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 3012	. . . . . . . . . . . . . . . . . . . . 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 4686	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘0), (𝑃‘1)} = {(𝑃‘2), (𝑃‘1)})
43		prcom 4685	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {(𝑃‘2), (𝑃‘1)} = {(𝑃‘1), (𝑃‘2)}
44	42, 43	eqtrdi 2807	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘0), (𝑃‘1)} = {(𝑃‘1), (𝑃‘2)})
45	44	necon3i 2983	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)} → (𝑃‘0) ≠ (𝑃‘2))
46	41, 45	biimtrdi 255	. . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 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 244	. . . . . . . . . . 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 413	. . . . . . . . . 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 483	. . . . . . . . 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 242	. . . . . . . 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 1140	. . . . . 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 417	. . . . 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 242	. . 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 409	1 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136   ≠ wne 2951  ∀wral 3070  Vcvv 3448  {cpr 4578   class class class wbr 5094  dom cdm 5640  ⟶wf 6506  –1-1→wf1 6507  –1-1-onto→wf1o 6509  ‘cfv 6510  (class class class)co 7385  0cc0 11063  1c1 11064   + caddc 11066  2c2 12262  ...cfz 13502  ..^cfzo 13649  ♯chash 14333  Vtxcvtx 29136  iEdgciedg 29137  Edgcedg 29187  UPGraphcupgr 29220  USGraphcusgr 29289  Trailsctrls 29828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707  ax-cnex 11119  ax-resscn 11120  ax-1cn 11121  ax-icn 11122  ax-addcl 11123  ax-addrcl 11124  ax-mulcl 11125  ax-mulrcl 11126  ax-mulcom 11127  ax-addass 11128  ax-mulass 11129  ax-distr 11130  ax-i2m1 11131  ax-1ne0 11132  ax-1rid 11133  ax-rnegex 11134  ax-rrecex 11135  ax-cnre 11136  ax-pre-lttri 11137  ax-pre-lttrn 11138  ax-pre-ltadd 11139  ax-pre-mulgt0 11140

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-ifp 1072  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-nel 3056  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4900  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-1o 8425  df-2o 8426  df-oadd 8429  df-er 8666  df-map 8798  df-pm 8799  df-en 8917  df-dom 8918  df-sdom 8919  df-fin 8920  df-dju 9849  df-card 9887  df-pnf 11208  df-mnf 11209  df-xr 11210  df-ltxr 11211  df-le 11212  df-sub 11406  df-neg 11407  df-nn 12201  df-2 12270  df-n0 12472  df-xnn0 12545  df-z 12559  df-uz 12830  df-fz 13503  df-fzo 13650  df-hash 14334  df-word 14517  df-edg 29188  df-uhgr 29198  df-upgr 29222  df-uspgr 29290  df-usgr 29291  df-wlks 29739  df-trls 29830

This theorem is referenced by:  usgr2trlspth  29900  usgr2trlncrct  29945