Theorem 2spotiundisj 26589

Description: All simple paths of length 2 as ordered triple from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 5-Mar-2018.)

Assertion

Ref	Expression
2spotiundisj	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏))

Distinct variable groups:   𝐸,𝑎,𝑏   𝑉,𝑎,𝑏   𝑋,𝑎,𝑏   𝑌,𝑎,𝑏

Proof of Theorem 2spotiundisj

Dummy variables 𝑐 𝑑 𝑓 𝑚 𝑛 𝑝 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 399	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅))
2	1	a1d 25	. . . 4 ⊢ (𝑎 = 𝑐 → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅)))
3		eliun 4460	. . . . . . . . . 10 ⊢ (𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ↔ ∃𝑏 ∈ (𝑉 ∖ {𝑎})𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏))
4		simpl 472	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌))
5	4	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) → (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌))
6	5	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌))
7		simprrl 800	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) → 𝑎 ∈ 𝑉)
8	7	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → 𝑎 ∈ 𝑉)
9		eldifi 3694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑏 ∈ 𝑉)
10	9	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → 𝑏 ∈ 𝑉)
11		el2spthonot 26397	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ↔ ∃𝑚 ∈ 𝑉 (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))))
12	6, 8, 10, 11	syl12anc 1316	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ↔ ∃𝑚 ∈ 𝑉 (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))))
13		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑎 ∈ V
14		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑚 ∈ V
15		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑏 ∈ V
16	13, 14, 15	otth 4879	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩ ↔ (𝑎 = 𝑐 ∧ 𝑚 = 𝑛 ∧ 𝑏 = 𝑑))
17	16	simp1bi 1069	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩ → 𝑎 = 𝑐)
18	17	con3i 149	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (¬ 𝑎 = 𝑐 → ¬ ⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩)
19	18	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚 ∈ 𝑉) → ¬ ⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩)
20		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ → (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ↔ ⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩))
21	20	notbid 307	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ↔ ¬ ⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩))
22	19, 21	syl5ibr 235	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ → ((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚 ∈ 𝑉) → ¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩))
23	22	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2)))) → ((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚 ∈ 𝑉) → ¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩))
24	23	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚 ∈ 𝑉) ∧ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))) → ¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩)
25	24	orcd 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚 ∈ 𝑉) ∧ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))) → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))
26	25	a1d 25	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚 ∈ 𝑉) ∧ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))) → (𝑛 ∈ 𝑉 → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))
27	26	a1d 25	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚 ∈ 𝑉) ∧ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → (𝑛 ∈ 𝑉 → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))))
28	27	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚 ∈ 𝑉) → ((𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2)))) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → (𝑛 ∈ 𝑉 → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))))
29	28	rexlimdva 3013	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (∃𝑚 ∈ 𝑉 (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2)))) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → (𝑛 ∈ 𝑉 → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))))
30	12, 29	sylbid 229	. . . . . . . . . . . . . . . . . 18 ⊢ (((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → (𝑛 ∈ 𝑉 → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))))
31	30	imp41 617	. . . . . . . . . . . . . . . . 17 ⊢ ((((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) ∧ 𝑛 ∈ 𝑉) → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))
32		ianor 508	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))) ↔ (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))
33	31, 32	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ ((((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) ∧ 𝑛 ∈ 𝑉) → ¬ (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))
34	33	nrexdv 2984	. . . . . . . . . . . . . . 15 ⊢ (((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → ¬ ∃𝑛 ∈ 𝑉 (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))
35	4	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌))
36		simprr 792	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑐 ∈ 𝑉)
37		eldifi 3694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ∈ (𝑉 ∖ {𝑐}) → 𝑑 ∈ 𝑉)
38	36, 37	anim12i 588	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
39	35, 38	jca 553	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)))
40	39	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))))
41	40	ad3antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ ((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))))
42	41	imp 444	. . . . . . . . . . . . . . . 16 ⊢ (((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)))
43		el2spthonot 26397	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑡 ∈ (𝑐(𝑉 2SPathOnOt 𝐸)𝑑) ↔ ∃𝑛 ∈ 𝑉 (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))
44	42, 43	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → (𝑡 ∈ (𝑐(𝑉 2SPathOnOt 𝐸)𝑑) ↔ ∃𝑛 ∈ 𝑉 (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))
45	34, 44	mtbird 314	. . . . . . . . . . . . . 14 ⊢ (((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → ¬ 𝑡 ∈ (𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
46	45	nrexdv 2984	. . . . . . . . . . . . 13 ⊢ ((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) → ¬ ∃𝑑 ∈ (𝑉 ∖ {𝑐})𝑡 ∈ (𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
47		eliun 4460	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑) ↔ ∃𝑑 ∈ (𝑉 ∖ {𝑐})𝑡 ∈ (𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
48	46, 47	sylnibr 318	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) → ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
49	48	ex 449	. . . . . . . . . . 11 ⊢ (((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) → ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑)))
50	49	rexlimdva 3013	. . . . . . . . . 10 ⊢ ((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) → (∃𝑏 ∈ (𝑉 ∖ {𝑎})𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) → ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑)))
51	3, 50	syl5bi 231	. . . . . . . . 9 ⊢ ((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) → (𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) → ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑)))
52	51	ralrimiv 2948	. . . . . . . 8 ⊢ ((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) → ∀𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
53		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → (𝑐(𝑉 2SPathOnOt 𝐸)𝑏) = (𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
54	53	cbviunv 4495	. . . . . . . . . . 11 ⊢ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏) = ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑)
55	54	eleq2i 2680	. . . . . . . . . 10 ⊢ (𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏) ↔ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
56	55	notbii 309	. . . . . . . . 9 ⊢ (¬ 𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏) ↔ ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
57	56	ralbii 2963	. . . . . . . 8 ⊢ (∀𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ¬ 𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏) ↔ ∀𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
58	52, 57	sylibr 223	. . . . . . 7 ⊢ ((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) → ∀𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ¬ 𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏))
59		disj 3969	. . . . . . 7 ⊢ ((∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅ ↔ ∀𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ¬ 𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏))
60	58, 59	sylibr 223	. . . . . 6 ⊢ ((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) → (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅)
61	60	olcd 407	. . . . 5 ⊢ ((¬ 𝑎 = 𝑐 ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) → (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅))
62	61	ex 449	. . . 4 ⊢ (¬ 𝑎 = 𝑐 → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅)))
63	2, 62	pm2.61i 175	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅))
64	63	ralrimivva 2954	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ 𝑉 (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅))
65		sneq 4135	. . . . 5 ⊢ (𝑎 = 𝑐 → {𝑎} = {𝑐})
66	65	difeq2d 3690	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝑐}))
67		oveq1 6556	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) = (𝑐(𝑉 2SPathOnOt 𝐸)𝑏))
68	66, 67	iuneq12d 4482	. . 3 ⊢ (𝑎 = 𝑐 → ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) = ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏))
69	68	disjor 4567	. 2 ⊢ (Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ↔ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ 𝑉 (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅))
70	64, 69	sylibr 223	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ∩ cin 3539  ∅c0 3874  {csn 4125  ⟨cotp 4133  ∪ ciun 4455  Disj wdisj 4553   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   SPaths cspath 26029   2SPathOnOt c2pthonot 26384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2spthonot 26387

This theorem is referenced by:  frghash2spot  26590