Theorem usgr2pth0 26661

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 26660	. 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 3092	. . . . . . . . 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 4205	. . . . . . . . 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 266	. . . . . . . 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 3041	. . . . . . 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 3099	. . . . . . 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 1039	. . . . . . . . . . 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 681	. . . . . . . . . . 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 681	. . . . . . . . . . . 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 681	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ↔ (𝑦 ≠ 𝑥 ∧ (𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥)))
13		ancom 466	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≠ 𝑥 ∧ (𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥)) ↔ ((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ∧ 𝑦 ≠ 𝑥))
14		necom 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ 𝑧 ↔ 𝑧 ≠ 𝑦)
15	14	anbi2ci 732	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ↔ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦))
16	15	anbi1i 731	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ∧ 𝑦 ≠ 𝑥) ↔ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ 𝑦 ≠ 𝑥))
17	12, 13, 16	3bitri 286	. . . . . . . . . . . . 13 ⊢ (((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ↔ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ 𝑦 ≠ 𝑥))
18	17	anbi1i 731	. . . . . . . . . . . 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 1039	. . . . . . . . . . . 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 292	. . . . . . . . . . 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 288	. . . . . . . . . 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 3041	. . . . . . . . 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 4205	. . . . . . . . 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 3092	. . . . . . . . 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 288	. . . . . . . 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 3041	. . . . . . 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 286	. . . . . 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 4323	. . . . . 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 4323	. . . . . 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 292	. . . . 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 3049	. . 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 1405	. 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 268	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∖ cdif 3571  {csn 4177  {cpr 4179   class class class wbr 4653  dom cdm 5114  –1-1→wf1 5885  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937  2c2 11070  ...cfz 12326  ..^cfzo 12465  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   USGraph cusgr 26044  Pathscpths 26608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-pthson 26614  df-spthson 26615

This theorem is referenced by: (None)