Theorem usgr2pth0 29695

Description: In a simply graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.)

Hypotheses

Ref	Expression
usgr2pthlem.v	⊢ 𝑉 = (Vtx‘𝐺)
usgr2pthlem.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
usgr2pth0	⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))

Distinct variable groups:   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐼,𝑦,𝑧   𝑥,𝑃,𝑦,𝑧   𝑥,𝑉,𝑦,𝑧

Proof of Theorem usgr2pth0

Step	Hyp	Ref	Expression
1		usgr2pthlem.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		usgr2pthlem.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	usgr2pth 29694	. 2 ⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑥})∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
4		r19.42v 3169	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ (𝑧 ≠ 𝑥 ∧ ∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))))
5		rexdifpr 4623	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ ∃𝑦 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧 ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
6	4, 5	bitr3i 277	. . . . . . . 8 ⊢ ((𝑧 ≠ 𝑥 ∧ ∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ ∃𝑦 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧 ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
7	6	rexbii 3076	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝑉 (𝑧 ≠ 𝑥 ∧ ∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ ∃𝑧 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧 ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
8		rexcom 3266	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧 ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))) ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧 ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
9		df-3an 1088	. . . . . . . . . . 11 ⊢ ((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧 ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))) ↔ ((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
10		anass 468	. . . . . . . . . . 11 ⊢ ((((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ ((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
11		anass 468	. . . . . . . . . . . 12 ⊢ ((((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ 𝑦 ≠ 𝑥) ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ (𝑦 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
12		anass 468	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ↔ (𝑦 ≠ 𝑥 ∧ (𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥)))
13		ancom 460	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≠ 𝑥 ∧ (𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥)) ↔ ((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ∧ 𝑦 ≠ 𝑥))
14		necom 2978	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ 𝑧 ↔ 𝑧 ≠ 𝑦)
15	14	anbi2ci 625	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ↔ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦))
16	15	anbi1i 624	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ∧ 𝑦 ≠ 𝑥) ↔ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ 𝑦 ≠ 𝑥))
17	12, 13, 16	3bitri 297	. . . . . . . . . . . . 13 ⊢ (((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ↔ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ 𝑦 ≠ 𝑥))
18	17	anbi1i 624	. . . . . . . . . . . 12 ⊢ ((((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ (((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ 𝑦 ≠ 𝑥) ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))))
19		df-3an 1088	. . . . . . . . . . . 12 ⊢ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ (𝑦 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))) ↔ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ (𝑦 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
20	11, 18, 19	3bitr4i 303	. . . . . . . . . . 11 ⊢ ((((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ (𝑦 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
21	9, 10, 20	3bitr2i 299	. . . . . . . . . 10 ⊢ ((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧 ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))) ↔ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ (𝑦 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
22	21	rexbii 3076	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧 ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))) ↔ ∃𝑧 ∈ 𝑉 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ (𝑦 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
23		rexdifpr 4623	. . . . . . . . 9 ⊢ (∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(𝑦 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ ∃𝑧 ∈ 𝑉 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ (𝑦 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
24		r19.42v 3169	. . . . . . . . 9 ⊢ (∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(𝑦 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ (𝑦 ≠ 𝑥 ∧ ∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))))
25	22, 23, 24	3bitr2i 299	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧 ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))) ↔ (𝑦 ≠ 𝑥 ∧ ∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))))
26	25	rexbii 3076	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧 ∧ (𝑧 ≠ 𝑥 ∧ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))) ↔ ∃𝑦 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ ∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))))
27	7, 8, 26	3bitri 297	. . . . . 6 ⊢ (∃𝑧 ∈ 𝑉 (𝑧 ≠ 𝑥 ∧ ∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ ∃𝑦 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ ∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))))
28		rexdifsn 4758	. . . . . 6 ⊢ (∃𝑧 ∈ (𝑉 ∖ {𝑥})∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})) ↔ ∃𝑧 ∈ 𝑉 (𝑧 ≠ 𝑥 ∧ ∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))))
29		rexdifsn 4758	. . . . . 6 ⊢ (∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})) ↔ ∃𝑦 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ ∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))))
30	27, 28, 29	3bitr4i 303	. . . . 5 ⊢ (∃𝑧 ∈ (𝑉 ∖ {𝑥})∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})) ↔ ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))
31	30	a1i 11	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ 𝑉) → (∃𝑧 ∈ (𝑉 ∖ {𝑥})∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})) ↔ ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))))
32	31	rexbidva 3155	. . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑥 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑥})∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))))
33	32	3anbi3d 1444	. 2 ⊢ (𝐺 ∈ USGraph → ((𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑥})∃𝑦 ∈ (𝑉 ∖ {𝑥, 𝑧})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦}))) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
34	3, 33	bitrd 279	1 ⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ∖ cdif 3911  {csn 4589  {cpr 4591   class class class wbr 5107  dom cdm 5638  –1-1→wf1 6508  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069  2c2 12241  ...cfz 13468  ..^cfzo 13615  ♯chash 14295  Vtxcvtx 28923  iEdgciedg 28924  USGraphcusgr 29076  Pathscpths 29640

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-xnn0 12516  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-hash 14296  df-word 14479  df-concat 14536  df-s1 14561  df-s2 14814  df-s3 14815  df-edg 28975  df-uhgr 28985  df-upgr 29009  df-umgr 29010  df-uspgr 29077  df-usgr 29078  df-wlks 29527  df-wlkson 29528  df-trls 29620  df-trlson 29621  df-pths 29644  df-spths 29645  df-pthson 29646  df-spthson 29647

This theorem is referenced by: (None)