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Theorem 1wlkv0 40859
 Description: If there is a walk in the null graph (a class without vertices), it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
Assertion
Ref Expression
1wlkv0 (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (1Walks‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))

Proof of Theorem 1wlkv0
StepHypRef Expression
1 1wlkcpr 40833 . . 3 (𝑊 ∈ (1Walks‘𝐺) ↔ (1st𝑊)(1Walks‘𝐺)(2nd𝑊))
2 eqid 2610 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2610 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 32m1wlk 40818 . . . 4 ((1st𝑊)(1Walks‘𝐺)(2nd𝑊) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)))
5 feq3 5941 . . . . . . . 8 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ (2nd𝑊):(0...(#‘(1st𝑊)))⟶∅))
6 f00 6000 . . . . . . . 8 ((2nd𝑊):(0...(#‘(1st𝑊)))⟶∅ ↔ ((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅))
75, 6syl6bb 275 . . . . . . 7 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ ((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅)))
8 0z 11265 . . . . . . . . . . . . . 14 0 ∈ ℤ
9 nn0z 11277 . . . . . . . . . . . . . 14 ((#‘(1st𝑊)) ∈ ℕ0 → (#‘(1st𝑊)) ∈ ℤ)
10 fzn 12228 . . . . . . . . . . . . . 14 ((0 ∈ ℤ ∧ (#‘(1st𝑊)) ∈ ℤ) → ((#‘(1st𝑊)) < 0 ↔ (0...(#‘(1st𝑊))) = ∅))
118, 9, 10sylancr 694 . . . . . . . . . . . . 13 ((#‘(1st𝑊)) ∈ ℕ0 → ((#‘(1st𝑊)) < 0 ↔ (0...(#‘(1st𝑊))) = ∅))
12 nn0nlt0 11196 . . . . . . . . . . . . . 14 ((#‘(1st𝑊)) ∈ ℕ0 → ¬ (#‘(1st𝑊)) < 0)
1312pm2.21d 117 . . . . . . . . . . . . 13 ((#‘(1st𝑊)) ∈ ℕ0 → ((#‘(1st𝑊)) < 0 → (1st𝑊) = ∅))
1411, 13sylbird 249 . . . . . . . . . . . 12 ((#‘(1st𝑊)) ∈ ℕ0 → ((0...(#‘(1st𝑊))) = ∅ → (1st𝑊) = ∅))
1514com12 32 . . . . . . . . . . 11 ((0...(#‘(1st𝑊))) = ∅ → ((#‘(1st𝑊)) ∈ ℕ0 → (1st𝑊) = ∅))
1615adantl 481 . . . . . . . . . 10 (((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) → ((#‘(1st𝑊)) ∈ ℕ0 → (1st𝑊) = ∅))
17 lencl 13179 . . . . . . . . . 10 ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → (#‘(1st𝑊)) ∈ ℕ0)
1816, 17impel 484 . . . . . . . . 9 ((((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → (1st𝑊) = ∅)
19 simpll 786 . . . . . . . . 9 ((((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → (2nd𝑊) = ∅)
2018, 19jca 553 . . . . . . . 8 ((((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
2120ex 449 . . . . . . 7 (((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
227, 21syl6bi 242 . . . . . 6 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))))
2322com23 84 . . . . 5 ((Vtx‘𝐺) = ∅ → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))))
2423impd 446 . . . 4 ((Vtx‘𝐺) = ∅ → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
254, 24syl5 33 . . 3 ((Vtx‘𝐺) = ∅ → ((1st𝑊)(1Walks‘𝐺)(2nd𝑊) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
261, 25syl5bi 231 . 2 ((Vtx‘𝐺) = ∅ → (𝑊 ∈ (1Walks‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
2726imp 444 1 (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (1Walks‘𝐺)) → ((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  Vtxcvtx 25673  iEdgciedg 25674  1Walksc1wlks 40796 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-ifp 1007  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-1wlks 40800 This theorem is referenced by:  g01wlk0  40860
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