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Theorem wlkntrl 26092
 Description: A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one edge to the other is a walk, but not a trail. Notice that ⟨𝑉, 𝐸⟩ is a simple graph (without loops) only if 𝑥 ≠ 𝑦. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
Hypotheses
Ref Expression
wlkntrl.v 𝑉 = {𝑥, 𝑦}
wlkntrl.e 𝐸 = {⟨0, {𝑥, 𝑦}⟩}
wlkntrl.f 𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}
wlkntrl.p 𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}
Assertion
Ref Expression
wlkntrl (𝐹(𝑉 Walks 𝐸)𝑃 ∧ ¬ 𝐹(𝑉 Trails 𝐸)𝑃)
Distinct variable group:   𝑥,𝐹,𝑦
Allowed substitution hints:   𝑃(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem wlkntrl
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 wlkntrl.v . . . . . 6 𝑉 = {𝑥, 𝑦}
2 prex 4836 . . . . . 6 {𝑥, 𝑦} ∈ V
31, 2eqeltri 2684 . . . . 5 𝑉 ∈ V
4 wlkntrl.e . . . . . 6 𝐸 = {⟨0, {𝑥, 𝑦}⟩}
5 snex 4835 . . . . . 6 {⟨0, {𝑥, 𝑦}⟩} ∈ V
64, 5eqeltri 2684 . . . . 5 𝐸 ∈ V
73, 6pm3.2i 470 . . . 4 (𝑉 ∈ V ∧ 𝐸 ∈ V)
8 wlkntrl.f . . . . . 6 𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}
9 prex 4836 . . . . . 6 {⟨0, 0⟩, ⟨1, 0⟩} ∈ V
108, 9eqeltri 2684 . . . . 5 𝐹 ∈ V
11 wlkntrl.p . . . . . 6 𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}
12 tpex 6855 . . . . . 6 {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩} ∈ V
1311, 12eqeltri 2684 . . . . 5 𝑃 ∈ V
1410, 13pm3.2i 470 . . . 4 (𝐹 ∈ V ∧ 𝑃 ∈ V)
157, 14pm3.2i 470 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))
161, 4, 8, 11wlkntrllem1 26089 . . . . 5 𝐹 ∈ Word dom 𝐸
17 vex 3176 . . . . . . . . . . 11 𝑥 ∈ V
1817prid1 4241 . . . . . . . . . 10 𝑥 ∈ {𝑥, 𝑦}
19 vex 3176 . . . . . . . . . . 11 𝑦 ∈ V
2019prid2 4242 . . . . . . . . . 10 𝑦 ∈ {𝑥, 𝑦}
2118, 20, 183pm3.2i 1232 . . . . . . . . 9 (𝑥 ∈ {𝑥, 𝑦} ∧ 𝑦 ∈ {𝑥, 𝑦} ∧ 𝑥 ∈ {𝑥, 𝑦})
22 eleq2 2677 . . . . . . . . . 10 (𝑉 = {𝑥, 𝑦} → (𝑥𝑉𝑥 ∈ {𝑥, 𝑦}))
23 eleq2 2677 . . . . . . . . . 10 (𝑉 = {𝑥, 𝑦} → (𝑦𝑉𝑦 ∈ {𝑥, 𝑦}))
2422, 23, 223anbi123d 1391 . . . . . . . . 9 (𝑉 = {𝑥, 𝑦} → ((𝑥𝑉𝑦𝑉𝑥𝑉) ↔ (𝑥 ∈ {𝑥, 𝑦} ∧ 𝑦 ∈ {𝑥, 𝑦} ∧ 𝑥 ∈ {𝑥, 𝑦})))
2521, 24mpbiri 247 . . . . . . . 8 (𝑉 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉𝑥𝑉))
261, 25ax-mp 5 . . . . . . 7 (𝑥𝑉𝑦𝑉𝑥𝑉)
27112trllemG 26088 . . . . . . 7 ((𝑥𝑉𝑦𝑉𝑥𝑉) → 𝑃:(0...2)⟶𝑉)
2826, 27ax-mp 5 . . . . . 6 𝑃:(0...2)⟶𝑉
29 0z 11265 . . . . . . . . . 10 0 ∈ ℤ
3029, 29pm3.2i 470 . . . . . . . . 9 (0 ∈ ℤ ∧ 0 ∈ ℤ)
3130, 82trllemA 26080 . . . . . . . 8 (#‘𝐹) = 2
3231oveq2i 6560 . . . . . . 7 (0...(#‘𝐹)) = (0...2)
3332feq2i 5950 . . . . . 6 (𝑃:(0...(#‘𝐹))⟶𝑉𝑃:(0...2)⟶𝑉)
3428, 33mpbir 220 . . . . 5 𝑃:(0...(#‘𝐹))⟶𝑉
351, 4, 8, 11wlkntrllem2 26090 . . . . 5 𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}
3616, 34, 353pm3.2i 1232 . . . 4 (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
37 iswlk 26048 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Walks 𝐸)𝑃 ↔ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
3836, 37mpbiri 247 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → 𝐹(𝑉 Walks 𝐸)𝑃)
3915, 38ax-mp 5 . 2 𝐹(𝑉 Walks 𝐸)𝑃
401, 4, 8, 11wlkntrllem3 26091 . . 3 ¬ Fun 𝐹
41 istrl 26067 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
4215, 41ax-mp 5 . . . 4 (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
43 notnot 135 . . . . . 6 (Fun 𝐹 → ¬ ¬ Fun 𝐹)
4443adantl 481 . . . . 5 ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → ¬ ¬ Fun 𝐹)
45443ad2ant1 1075 . . . 4 (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ¬ ¬ Fun 𝐹)
4642, 45sylbi 206 . . 3 (𝐹(𝑉 Trails 𝐸)𝑃 → ¬ ¬ Fun 𝐹)
4740, 46mt2 190 . 2 ¬ 𝐹(𝑉 Trails 𝐸)𝑃
4839, 47pm3.2i 470 1 (𝐹(𝑉 Walks 𝐸)𝑃 ∧ ¬ 𝐹(𝑉 Trails 𝐸)𝑃)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  {csn 4125  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   Walks cwalk 26026   Trails ctrail 26027 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-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-cda 8873  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 This theorem is referenced by: (None)
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