Theorem usgr2pth0 28034

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 28033	. 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 3276	. . . . . . . . 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 4591	. . . . . . . . 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 276	. . . . . . . 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 3177	. . . . . . 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 3281	. . . . . . 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 1087	. . . . . . . . . . 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 2996	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ 𝑧 ↔ 𝑧 ≠ 𝑦)
15	14	anbi2ci 624	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ↔ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦))
16	15	anbi1i 623	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ≠ 𝑧 ∧ 𝑧 ≠ 𝑥) ∧ 𝑦 ≠ 𝑥) ↔ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ 𝑦 ≠ 𝑥))
17	12, 13, 16	3bitri 296	. . . . . . . . . . . . 13 ⊢ (((𝑦 ≠ 𝑥 ∧ 𝑦 ≠ 𝑧) ∧ 𝑧 ≠ 𝑥) ↔ ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦) ∧ 𝑦 ≠ 𝑥))
18	17	anbi1i 623	. . . . . . . . . . . 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 1087	. . . . . . . . . . . 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 302	. . . . . . . . . . 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 298	. . . . . . . . . 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 3177	. . . . . . . . 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 4591	. . . . . . . . 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 3276	. . . . . . . . 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 298	. . . . . . . 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 3177	. . . . . . 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 296	. . . . . 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 4724	. . . . . 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 4724	. . . . . 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 302	. . . . 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 3224	. . 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 1440	. 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 278	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 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   ∖ cdif 3880  {csn 4558  {cpr 4560   class class class wbr 5070  dom cdm 5580  –1-1→wf1 6415  ‘cfv 6418  (class class class)co 7255  0cc0 10802  1c1 10803  2c2 11958  ...cfz 13168  ..^cfzo 13311  ♯chash 13972  Vtxcvtx 27269  iEdgciedg 27270  USGraphcusgr 27422  Pathscpths 27981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-s2 14489  df-s3 14490  df-edg 27321  df-uhgr 27331  df-upgr 27355  df-umgr 27356  df-uspgr 27423  df-usgr 27424  df-wlks 27869  df-wlkson 27870  df-trls 27962  df-trlson 27963  df-pths 27985  df-spths 27986  df-pthson 27987  df-spthson 27988

This theorem is referenced by: (None)