Theorem usgr2pth0 30051

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 30050	. 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 3203	. . . . . . . . 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 4627	. . . . . . . . 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 280	. . . . . . . 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 3118	. . . . . . 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 3300	. . . . . . 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 1103	. . . . . . . . . . 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 473	. . . . . . . . . . 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 473	. . . . . . . . . . . 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 473	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ↔ (𝑦 ≠ 𝑥 ∧ (𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥)))
13		ancom 465	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≠ 𝑥 ∧ (𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥)) ↔ ((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ∧ 𝑦 ≠ 𝑥))
14		necom 3017	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ 𝑧 ↔ 𝑧 ≠ 𝑦)
15	14	anbi2ci 636	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ↔ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦))
16	15	anbi1i 635	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ∧ 𝑦 ≠ 𝑥) ↔ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ 𝑦 ≠ 𝑥))
17	12, 13, 16	3bitri 300	. . . . . . . . . . . . 13 ⊢ (((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ↔ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ 𝑦 ≠ 𝑥))
18	17	anbi1i 635	. . . . . . . . . . . 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 1103	. . . . . . . . . . . 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 306	. . . . . . . . . . 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 302	. . . . . . . . . 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 3118	. . . . . . . . 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 4627	. . . . . . . . 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 3203	. . . . . . . . 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 302	. . . . . . . 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 3118	. . . . . . 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 300	. . . . . 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 4763	. . . . . 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 4763	. . . . . 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 306	. . . . 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 3193	. . 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 1468	. 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 282	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095   ∖ cdif 3910  {csn 4591  {cpr 4593   class class class wbr 5110  dom cdm 5659  –1-1→wf1 6530  ‘cfv 6533  (class class class)co 7408  0cc0 11096  1c1 11097  2c2 12291  ...cfz 13531  ..^cfzo 13678  ♯chash 14362  Vtxcvtx 29283  iEdgciedg 29284  USGraphcusgr 29436  Pathscpths 29996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-er 8690  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9883  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-n0 12501  df-xnn0 12574  df-z 12588  df-uz 12859  df-fz 13532  df-fzo 13679  df-hash 14363  df-word 14547  df-concat 14604  df-s1 14630  df-s2 14881  df-s3 14882  df-edg 29335  df-uhgr 29345  df-upgr 29369  df-umgr 29370  df-uspgr 29437  df-usgr 29438  df-wlks 29886  df-wlkson 29887  df-trls 29977  df-trlson 29978  df-pths 30000  df-spths 30001  df-pthson 30002  df-spthson 30003

This theorem is referenced by: (None)