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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 ((𝑉𝑋𝐸𝑌) → Disj 𝑎𝑉 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.
StepHypRef 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)) = ∅))
21a1d 25 . . . 4 (𝑎 = 𝑐 → (((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉)) → (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 (((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉)) → (𝑉𝑋𝐸𝑌))
54adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) → (𝑉𝑋𝐸𝑌))
65adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑉𝑋𝐸𝑌))
7 simprrl 800 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) → 𝑎𝑉)
87adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → 𝑎𝑉)
9 eldifi 3694 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑏𝑉)
109adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → 𝑏𝑉)
11 el2spthonot 26397 . . . . . . . . . . . . . . . . . . . 20 (((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑏𝑉)) → (𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ↔ ∃𝑚𝑉 (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))))
126, 8, 10, 11syl12anc 1316 . . . . . . . . . . . . . . . . . . 19 (((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ↔ ∃𝑚𝑉 (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓𝑝(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
1613, 14, 15otth 4879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩ ↔ (𝑎 = 𝑐𝑚 = 𝑛𝑏 = 𝑑))
1716simp1bi 1069 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩ → 𝑎 = 𝑐)
1817con3i 149 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑎 = 𝑐 → ¬ ⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩)
1918ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → ¬ ⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩)
20 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ → (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ↔ ⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩))
2120notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ↔ ¬ ⟨𝑎, 𝑚, 𝑏⟩ = ⟨𝑐, 𝑛, 𝑑⟩))
2219, 21syl5ibr 235 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ → ((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → ¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩))
2322adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2)))) → ((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → ¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩))
2423impcom 445 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))) → ¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩)
2524orcd 406 . . . . . . . . . . . . . . . . . . . . . . 23 (((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))) → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))
2625a1d 25 . . . . . . . . . . . . . . . . . . . . . 22 (((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))) → (𝑛𝑉 → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))
2726a1d 25 . . . . . . . . . . . . . . . . . . . . 21 (((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2))))) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → (𝑛𝑉 → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))))
2827ex 449 . . . . . . . . . . . . . . . . . . . 20 ((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → ((𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2)))) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → (𝑛𝑉 → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))))
2928rexlimdva 3013 . . . . . . . . . . . . . . . . . . 19 (((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (∃𝑚𝑉 (𝑡 = ⟨𝑎, 𝑚, 𝑏⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑚 = (𝑝‘1) ∧ 𝑏 = (𝑝‘2)))) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → (𝑛𝑉 → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))))
3012, 29sylbid 229 . . . . . . . . . . . . . . . . . 18 (((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → (𝑛𝑉 → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))))
3130imp41 617 . . . . . . . . . . . . . . . . 17 ((((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) ∧ 𝑛𝑉) → (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))
32 ianor 508 . . . . . . . . . . . . . . . . 17 (¬ (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))) ↔ (¬ 𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∨ ¬ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))
3331, 32sylibr 223 . . . . . . . . . . . . . . . 16 ((((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) ∧ 𝑛𝑉) → ¬ (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))
3433nrexdv 2984 . . . . . . . . . . . . . . 15 (((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → ¬ ∃𝑛𝑉 (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2)))))
354adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → (𝑉𝑋𝐸𝑌))
36 simprr 792 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉)) → 𝑐𝑉)
37 eldifi 3694 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ∈ (𝑉 ∖ {𝑐}) → 𝑑𝑉)
3836, 37anim12i 588 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → (𝑐𝑉𝑑𝑉))
3935, 38jca 553 . . . . . . . . . . . . . . . . . . 19 ((((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → ((𝑉𝑋𝐸𝑌) ∧ (𝑐𝑉𝑑𝑉)))
4039ex 449 . . . . . . . . . . . . . . . . . 18 (((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉)) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → ((𝑉𝑋𝐸𝑌) ∧ (𝑐𝑉𝑑𝑉))))
4140ad3antlr 763 . . . . . . . . . . . . . . . . 17 ((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) → (𝑑 ∈ (𝑉 ∖ {𝑐}) → ((𝑉𝑋𝐸𝑌) ∧ (𝑐𝑉𝑑𝑉))))
4241imp 444 . . . . . . . . . . . . . . . 16 (((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → ((𝑉𝑋𝐸𝑌) ∧ (𝑐𝑉𝑑𝑉)))
43 el2spthonot 26397 . . . . . . . . . . . . . . . 16 (((𝑉𝑋𝐸𝑌) ∧ (𝑐𝑉𝑑𝑉)) → (𝑡 ∈ (𝑐(𝑉 2SPathOnOt 𝐸)𝑑) ↔ ∃𝑛𝑉 (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))
4442, 43syl 17 . . . . . . . . . . . . . . 15 (((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → (𝑡 ∈ (𝑐(𝑉 2SPathOnOt 𝐸)𝑑) ↔ ∃𝑛𝑉 (𝑡 = ⟨𝑐, 𝑛, 𝑑⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑐 = (𝑝‘0) ∧ 𝑛 = (𝑝‘1) ∧ 𝑑 = (𝑝‘2))))))
4534, 44mtbird 314 . . . . . . . . . . . . . 14 (((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → ¬ 𝑡 ∈ (𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
4645nrexdv 2984 . . . . . . . . . . . . 13 ((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 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))
4846, 47sylnibr 318 . . . . . . . . . . . 12 ((((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) → ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
4948ex 449 . . . . . . . . . . 11 (((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) → ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑)))
5049rexlimdva 3013 . . . . . . . . . 10 ((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) → (∃𝑏 ∈ (𝑉 ∖ {𝑎})𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) → ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑)))
513, 50syl5bi 231 . . . . . . . . 9 ((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) → (𝑡 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) → ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑)))
5251ralrimiv 2948 . . . . . . . 8 ((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) → ∀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))
5453cbviunv 4495 . . . . . . . . . . 11 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏) = 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑)
5554eleq2i 2680 . . . . . . . . . 10 (𝑡 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏) ↔ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
5655notbii 309 . . . . . . . . 9 𝑡 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏) ↔ ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑑))
5756ralbii 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))
5852, 57sylibr 223 . . . . . . 7 ((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) → ∀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))
6058, 59sylibr 223 . . . . . 6 ((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) → ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅)
6160olcd 407 . . . . 5 ((¬ 𝑎 = 𝑐 ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉))) → (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅))
6261ex 449 . . . 4 𝑎 = 𝑐 → (((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉)) → (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅)))
632, 62pm2.61i 175 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑐𝑉)) → (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅))
6463ralrimivva 2954 . 2 ((𝑉𝑋𝐸𝑌) → ∀𝑎𝑉𝑐𝑉 (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})
6665difeq2d 3690 . . . 4 (𝑎 = 𝑐 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝑐}))
67 oveq1 6556 . . . 4 (𝑎 = 𝑐 → (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) = (𝑐(𝑉 2SPathOnOt 𝐸)𝑏))
6866, 67iuneq12d 4482 . . 3 (𝑎 = 𝑐 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) = 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏))
6968disjor 4567 . 2 (Disj 𝑎𝑉 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ↔ ∀𝑎𝑉𝑐𝑉 (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(𝑉 2SPathOnOt 𝐸)𝑏)) = ∅))
7064, 69sylibr 223 1 ((𝑉𝑋𝐸𝑌) → Disj 𝑎𝑉 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏))
 Colors of variables: wff setvar class 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
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