Theorem usgr2pth0 29012

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 29011	. 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 3191	. . . . . . . . 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 4661	. . . . . . . . 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 3095	. . . . . . 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 3288	. . . . . . 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 1090	. . . . . . . . . . 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 470	. . . . . . . . . . 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 470	. . . . . . . . . . . 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 470	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ↔ (𝑦 ≠ 𝑥 ∧ (𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥)))
13		ancom 462	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≠ 𝑥 ∧ (𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥)) ↔ ((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ∧ 𝑦 ≠ 𝑥))
14		necom 2995	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ 𝑧 ↔ 𝑧 ≠ 𝑦)
15	14	anbi2ci 626	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ↔ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦))
16	15	anbi1i 625	. . . . . . . . . . . . . 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 625	. . . . . . . . . . . 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 1090	. . . . . . . . . . . 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 3095	. . . . . . . . 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 4661	. . . . . . . . 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 3191	. . . . . . . . 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 3095	. . . . . . 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 4797	. . . . . 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 4797	. . . . . 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 3177	. . 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 1443	. 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 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∃wrex 3071   ∖ cdif 3945  {csn 4628  {cpr 4630   class class class wbr 5148  dom cdm 5676  –1-1→wf1 6538  ‘cfv 6541  (class class class)co 7406  0cc0 11107  1c1 11108  2c2 12264  ...cfz 13481  ..^cfzo 13624  ♯chash 14287  Vtxcvtx 28246  iEdgciedg 28247  USGraphcusgr 28399  Pathscpths 28959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-2o 8464  df-oadd 8467  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-dju 9893  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-xnn0 12542  df-z 12556  df-uz 12820  df-fz 13482  df-fzo 13625  df-hash 14288  df-word 14462  df-concat 14518  df-s1 14543  df-s2 14796  df-s3 14797  df-edg 28298  df-uhgr 28308  df-upgr 28332  df-umgr 28333  df-uspgr 28400  df-usgr 28401  df-wlks 28846  df-wlkson 28847  df-trls 28939  df-trlson 28940  df-pths 28963  df-spths 28964  df-pthson 28965  df-spthson 28966

This theorem is referenced by: (None)