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Theorem 2spontn0vne 26414
 Description: If the set of simple paths of length 2 between two vertices (in a graph) is not empty, the two vertices must be not equal. (Contributed by Alexander van der Vekens, 3-Mar-2018.)
Assertion
Ref Expression
2spontn0vne ((𝑋(𝑉 2SPathOnOt 𝐸)𝑌) ≠ ∅ → 𝑋𝑌)

Proof of Theorem 2spontn0vne
Dummy variables 𝑏 𝑓 𝑖 𝑝 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3890 . 2 ((𝑋(𝑉 2SPathOnOt 𝐸)𝑌) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ (𝑋(𝑉 2SPathOnOt 𝐸)𝑌))
2 2spthonot3v 26403 . . . 4 (𝑡 ∈ (𝑋(𝑉 2SPathOnOt 𝐸)𝑌) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉) ∧ 𝑡 ∈ ((𝑉 × 𝑉) × 𝑉)))
3 el2spthonot 26397 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) → (𝑡 ∈ (𝑋(𝑉 2SPathOnOt 𝐸)𝑌) ↔ ∃𝑏𝑉 (𝑡 = ⟨𝑋, 𝑏, 𝑌⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2))))))
4 vex 3176 . . . . . . . . . . . . . 14 𝑓 ∈ V
5 vex 3176 . . . . . . . . . . . . . 14 𝑝 ∈ V
6 isspth 26099 . . . . . . . . . . . . . . 15 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V)) → (𝑓(𝑉 SPaths 𝐸)𝑝 ↔ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun 𝑝)))
7 istrl2 26068 . . . . . . . . . . . . . . . 16 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V)) → (𝑓(𝑉 Trails 𝐸)𝑝 ↔ (𝑓:(0..^(#‘𝑓))–1-1→dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑝𝑖), (𝑝‘(𝑖 + 1))})))
87anbi1d 737 . . . . . . . . . . . . . . 15 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V)) → ((𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun 𝑝) ↔ ((𝑓:(0..^(#‘𝑓))–1-1→dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑝𝑖), (𝑝‘(𝑖 + 1))}) ∧ Fun 𝑝)))
96, 8bitrd 267 . . . . . . . . . . . . . 14 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V)) → (𝑓(𝑉 SPaths 𝐸)𝑝 ↔ ((𝑓:(0..^(#‘𝑓))–1-1→dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑝𝑖), (𝑝‘(𝑖 + 1))}) ∧ Fun 𝑝)))
104, 5, 9mpanr12 717 . . . . . . . . . . . . 13 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉 SPaths 𝐸)𝑝 ↔ ((𝑓:(0..^(#‘𝑓))–1-1→dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑝𝑖), (𝑝‘(𝑖 + 1))}) ∧ Fun 𝑝)))
1110adantr 480 . . . . . . . . . . . 12 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) → (𝑓(𝑉 SPaths 𝐸)𝑝 ↔ ((𝑓:(0..^(#‘𝑓))–1-1→dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑝𝑖), (𝑝‘(𝑖 + 1))}) ∧ Fun 𝑝)))
1211adantr 480 . . . . . . . . . . 11 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → (𝑓(𝑉 SPaths 𝐸)𝑝 ↔ ((𝑓:(0..^(#‘𝑓))–1-1→dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑝𝑖), (𝑝‘(𝑖 + 1))}) ∧ Fun 𝑝)))
13 df-f1 5809 . . . . . . . . . . . . . 14 (𝑝:(0...(#‘𝑓))–1-1𝑉 ↔ (𝑝:(0...(#‘𝑓))⟶𝑉 ∧ Fun 𝑝))
14 eqidd 2611 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑓) = 2 → 𝑝 = 𝑝)
15 oveq2 6557 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑓) = 2 → (0...(#‘𝑓)) = (0...2))
16 eqidd 2611 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑓) = 2 → 𝑉 = 𝑉)
1714, 15, 16f1eq123d 6044 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑓) = 2 → (𝑝:(0...(#‘𝑓))–1-1𝑉𝑝:(0...2)–1-1𝑉))
18 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . 23 (0...2) = (0...2)
1918f13idfv 12662 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝:(0...2)–1-1𝑉 ↔ (𝑝:(0...2)⟶𝑉 ∧ ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2))))
20 simpr2 1061 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝:(0...2)⟶𝑉 ∧ ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2))) → (𝑝‘0) ≠ (𝑝‘2))
2119, 20sylbi 206 . . . . . . . . . . . . . . . . . . . . 21 (𝑝:(0...2)–1-1𝑉 → (𝑝‘0) ≠ (𝑝‘2))
2221a1d 25 . . . . . . . . . . . . . . . . . . . 20 (𝑝:(0...2)–1-1𝑉 → ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → (𝑝‘0) ≠ (𝑝‘2)))
2317, 22syl6bi 242 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑓) = 2 → (𝑝:(0...(#‘𝑓))–1-1𝑉 → ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → (𝑝‘0) ≠ (𝑝‘2))))
2423com3l 87 . . . . . . . . . . . . . . . . . 18 (𝑝:(0...(#‘𝑓))–1-1𝑉 → ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → ((#‘𝑓) = 2 → (𝑝‘0) ≠ (𝑝‘2))))
2524imp31 447 . . . . . . . . . . . . . . . . 17 (((𝑝:(0...(#‘𝑓))–1-1𝑉 ∧ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉)) ∧ (#‘𝑓) = 2) → (𝑝‘0) ≠ (𝑝‘2))
2625adantr 480 . . . . . . . . . . . . . . . 16 ((((𝑝:(0...(#‘𝑓))–1-1𝑉 ∧ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉)) ∧ (#‘𝑓) = 2) ∧ (𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2))) → (𝑝‘0) ≠ (𝑝‘2))
27 simpl 472 . . . . . . . . . . . . . . . . . . 19 ((𝑋 = (𝑝‘0) ∧ 𝑌 = (𝑝‘2)) → 𝑋 = (𝑝‘0))
28 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((𝑋 = (𝑝‘0) ∧ 𝑌 = (𝑝‘2)) → 𝑌 = (𝑝‘2))
2927, 28neeq12d 2843 . . . . . . . . . . . . . . . . . 18 ((𝑋 = (𝑝‘0) ∧ 𝑌 = (𝑝‘2)) → (𝑋𝑌 ↔ (𝑝‘0) ≠ (𝑝‘2)))
30293adant2 1073 . . . . . . . . . . . . . . . . 17 ((𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2)) → (𝑋𝑌 ↔ (𝑝‘0) ≠ (𝑝‘2)))
3130adantl 481 . . . . . . . . . . . . . . . 16 ((((𝑝:(0...(#‘𝑓))–1-1𝑉 ∧ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉)) ∧ (#‘𝑓) = 2) ∧ (𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2))) → (𝑋𝑌 ↔ (𝑝‘0) ≠ (𝑝‘2)))
3226, 31mpbird 246 . . . . . . . . . . . . . . 15 ((((𝑝:(0...(#‘𝑓))–1-1𝑉 ∧ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉)) ∧ (#‘𝑓) = 2) ∧ (𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2))) → 𝑋𝑌)
3332exp41 636 . . . . . . . . . . . . . 14 (𝑝:(0...(#‘𝑓))–1-1𝑉 → ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → ((#‘𝑓) = 2 → ((𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2)) → 𝑋𝑌))))
3413, 33sylbir 224 . . . . . . . . . . . . 13 ((𝑝:(0...(#‘𝑓))⟶𝑉 ∧ Fun 𝑝) → ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → ((#‘𝑓) = 2 → ((𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2)) → 𝑋𝑌))))
35343ad2antl2 1217 . . . . . . . . . . . 12 (((𝑓:(0..^(#‘𝑓))–1-1→dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑝𝑖), (𝑝‘(𝑖 + 1))}) ∧ Fun 𝑝) → ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → ((#‘𝑓) = 2 → ((𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2)) → 𝑋𝑌))))
3635com12 32 . . . . . . . . . . 11 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → (((𝑓:(0..^(#‘𝑓))–1-1→dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑝𝑖), (𝑝‘(𝑖 + 1))}) ∧ Fun 𝑝) → ((#‘𝑓) = 2 → ((𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2)) → 𝑋𝑌))))
3712, 36sylbid 229 . . . . . . . . . 10 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → (𝑓(𝑉 SPaths 𝐸)𝑝 → ((#‘𝑓) = 2 → ((𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2)) → 𝑋𝑌))))
38373impd 1273 . . . . . . . . 9 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2))) → 𝑋𝑌))
3938exlimdvv 1849 . . . . . . . 8 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → (∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2))) → 𝑋𝑌))
4039adantld 482 . . . . . . 7 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏𝑉) → ((𝑡 = ⟨𝑋, 𝑏, 𝑌⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2)))) → 𝑋𝑌))
4140rexlimdva 3013 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) → (∃𝑏𝑉 (𝑡 = ⟨𝑋, 𝑏, 𝑌⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑋 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑌 = (𝑝‘2)))) → 𝑋𝑌))
423, 41sylbid 229 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉)) → (𝑡 ∈ (𝑋(𝑉 2SPathOnOt 𝐸)𝑌) → 𝑋𝑌))
43423adant3 1074 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑌𝑉) ∧ 𝑡 ∈ ((𝑉 × 𝑉) × 𝑉)) → (𝑡 ∈ (𝑋(𝑉 2SPathOnOt 𝐸)𝑌) → 𝑋𝑌))
442, 43mpcom 37 . . 3 (𝑡 ∈ (𝑋(𝑉 2SPathOnOt 𝐸)𝑌) → 𝑋𝑌)
4544exlimiv 1845 . 2 (∃𝑡 𝑡 ∈ (𝑋(𝑉 2SPathOnOt 𝐸)𝑌) → 𝑋𝑌)
461, 45sylbi 206 1 ((𝑋(𝑉 2SPathOnOt 𝐸)𝑌) ≠ ∅ → 𝑋𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ∅c0 3874  {cpr 4127  ⟨cotp 4133   class class class wbr 4583   × cxp 5036  ◡ccnv 5037  dom cdm 5038  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ...cfz 12197  ..^cfzo 12334  #chash 12979   Trails ctrail 26027   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-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-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:  usg2spthonot  26415  usg2spthonot0  26416  2spot2iun2spont  26418
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