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.

Step	Hyp	Ref	Expression
1		wlkntrl.v	. . . . . 6 ⊢ 𝑉 = {𝑥, 𝑦}
2		prex 4836	. . . . . 6 ⊢ {𝑥, 𝑦} ∈ V
3	1, 2	eqeltri 2684	. . . . 5 ⊢ 𝑉 ∈ V
4		wlkntrl.e	. . . . . 6 ⊢ 𝐸 = {⟨0, {𝑥, 𝑦}⟩}
5		snex 4835	. . . . . 6 ⊢ {⟨0, {𝑥, 𝑦}⟩} ∈ V
6	4, 5	eqeltri 2684	. . . . 5 ⊢ 𝐸 ∈ V
7	3, 6	pm3.2i 470	. . . 4 ⊢ (𝑉 ∈ V ∧ 𝐸 ∈ V)
8		wlkntrl.f	. . . . . 6 ⊢ 𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}
9		prex 4836	. . . . . 6 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} ∈ V
10	8, 9	eqeltri 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
13	11, 12	eqeltri 2684	. . . . 5 ⊢ 𝑃 ∈ V
14	10, 13	pm3.2i 470	. . . 4 ⊢ (𝐹 ∈ V ∧ 𝑃 ∈ V)
15	7, 14	pm3.2i 470	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))
16	1, 4, 8, 11	wlkntrllem1 26089	. . . . 5 ⊢ 𝐹 ∈ Word dom 𝐸
17		vex 3176	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
18	17	prid1 4241	. . . . . . . . . 10 ⊢ 𝑥 ∈ {𝑥, 𝑦}
19		vex 3176	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
20	19	prid2 4242	. . . . . . . . . 10 ⊢ 𝑦 ∈ {𝑥, 𝑦}
21	18, 20, 18	3pm3.2i 1232	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥, 𝑦} ∧ 𝑦 ∈ {𝑥, 𝑦} ∧ 𝑥 ∈ {𝑥, 𝑦})
22		eleq2 2677	. . . . . . . . . 10 ⊢ (𝑉 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ {𝑥, 𝑦}))
23		eleq2 2677	. . . . . . . . . 10 ⊢ (𝑉 = {𝑥, 𝑦} → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ {𝑥, 𝑦}))
24	22, 23, 22	3anbi123d 1391	. . . . . . . . 9 ⊢ (𝑉 = {𝑥, 𝑦} → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) ↔ (𝑥 ∈ {𝑥, 𝑦} ∧ 𝑦 ∈ {𝑥, 𝑦} ∧ 𝑥 ∈ {𝑥, 𝑦})))
25	21, 24	mpbiri 247	. . . . . . . 8 ⊢ (𝑉 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉))
26	1, 25	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)
27	11	2trllemG 26088	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑃:(0...2)⟶𝑉)
28	26, 27	ax-mp 5	. . . . . 6 ⊢ 𝑃:(0...2)⟶𝑉
29		0z 11265	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
30	29, 29	pm3.2i 470	. . . . . . . . 9 ⊢ (0 ∈ ℤ ∧ 0 ∈ ℤ)
31	30, 8	2trllemA 26080	. . . . . . . 8 ⊢ (#‘𝐹) = 2
32	31	oveq2i 6560	. . . . . . 7 ⊢ (0...(#‘𝐹)) = (0...2)
33	32	feq2i 5950	. . . . . 6 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉)
34	28, 33	mpbir 220	. . . . 5 ⊢ 𝑃:(0...(#‘𝐹))⟶𝑉
35	1, 4, 8, 11	wlkntrllem2 26090	. . . . 5 ⊢ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}
36	16, 34, 35	3pm3.2i 1232	. . . 4 ⊢ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
37		iswlk 26048	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Walks 𝐸)𝑃 ↔ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
38	36, 37	mpbiri 247	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → 𝐹(𝑉 Walks 𝐸)𝑃)
39	15, 38	ax-mp 5	. 2 ⊢ 𝐹(𝑉 Walks 𝐸)𝑃
40	1, 4, 8, 11	wlkntrllem3 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‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
42	15, 41	ax-mp 5	. . . 4 ⊢ (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun ◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
43		notnot 135	. . . . . 6 ⊢ (Fun ◡𝐹 → ¬ ¬ Fun ◡𝐹)
44	43	adantl 481	. . . . 5 ⊢ ((𝐹 ∈ Word dom 𝐸 ∧ Fun ◡𝐹) → ¬ ¬ Fun ◡𝐹)
45	44	3ad2ant1 1075	. . . 4 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ¬ ¬ Fun ◡𝐹)
46	42, 45	sylbi 206	. . 3 ⊢ (𝐹(𝑉 Trails 𝐸)𝑃 → ¬ ¬ Fun ◡𝐹)
47	40, 46	mt2 190	. 2 ⊢ ¬ 𝐹(𝑉 Trails 𝐸)𝑃
48	39, 47	pm3.2i 470	1 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ ¬ 𝐹(𝑉 Trails 𝐸)𝑃)

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)