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Theorem wlkv0 26288
 Description: If there is a walk in an empty graph, it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.)
Assertion
Ref Expression
wlkv0 (𝑊 ∈ (∅ Walks 𝐸) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))

Proof of Theorem wlkv0
StepHypRef Expression
1 wlkcpr 26057 . 2 (𝑊 ∈ (∅ Walks 𝐸) ↔ (1st𝑊)(∅ Walks 𝐸)(2nd𝑊))
2 2mwlk 26049 . . 3 ((1st𝑊)(∅ Walks 𝐸)(2nd𝑊) → ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶∅))
3 f00 6000 . . . 4 ((2nd𝑊):(0...(#‘(1st𝑊)))⟶∅ ↔ ((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅))
4 lencl 13179 . . . . . . 7 ((1st𝑊) ∈ Word dom 𝐸 → (#‘(1st𝑊)) ∈ ℕ0)
5 0z 11265 . . . . . . . . . 10 0 ∈ ℤ
6 nn0z 11277 . . . . . . . . . 10 ((#‘(1st𝑊)) ∈ ℕ0 → (#‘(1st𝑊)) ∈ ℤ)
7 fzn 12228 . . . . . . . . . 10 ((0 ∈ ℤ ∧ (#‘(1st𝑊)) ∈ ℤ) → ((#‘(1st𝑊)) < 0 ↔ (0...(#‘(1st𝑊))) = ∅))
85, 6, 7sylancr 694 . . . . . . . . 9 ((#‘(1st𝑊)) ∈ ℕ0 → ((#‘(1st𝑊)) < 0 ↔ (0...(#‘(1st𝑊))) = ∅))
9 nn0nlt0 11196 . . . . . . . . . 10 ((#‘(1st𝑊)) ∈ ℕ0 → ¬ (#‘(1st𝑊)) < 0)
109pm2.21d 117 . . . . . . . . 9 ((#‘(1st𝑊)) ∈ ℕ0 → ((#‘(1st𝑊)) < 0 → (1st𝑊) = ∅))
118, 10sylbird 249 . . . . . . . 8 ((#‘(1st𝑊)) ∈ ℕ0 → ((0...(#‘(1st𝑊))) = ∅ → (1st𝑊) = ∅))
1211adantld 482 . . . . . . 7 ((#‘(1st𝑊)) ∈ ℕ0 → (((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) → (1st𝑊) = ∅))
134, 12syl 17 . . . . . 6 ((1st𝑊) ∈ Word dom 𝐸 → (((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) → (1st𝑊) = ∅))
1413imp 444 . . . . 5 (((1st𝑊) ∈ Word dom 𝐸 ∧ ((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅)) → (1st𝑊) = ∅)
15 simprl 790 . . . . 5 (((1st𝑊) ∈ Word dom 𝐸 ∧ ((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅)) → (2nd𝑊) = ∅)
1614, 15jca 553 . . . 4 (((1st𝑊) ∈ Word dom 𝐸 ∧ ((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
173, 16sylan2b 491 . . 3 (((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶∅) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
182, 17syl 17 . 2 ((1st𝑊)(∅ Walks 𝐸)(2nd𝑊) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
191, 18sylbi 206 1 (𝑊 ∈ (∅ Walks 𝐸) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∅c0 3874   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815   < clt 9953  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197  #chash 12979  Word cword 13146   Walks cwalk 26026 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-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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036 This theorem is referenced by:  wlk0  26289
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