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Theorem wlkv0 28015
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
wlkv0 (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))

Proof of Theorem wlkv0
StepHypRef Expression
1 wlkcpr 27993 . . 3 (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))
2 eqid 2740 . . . . . 6 (iEdg‘𝐺) = (iEdg‘𝐺)
32wlkf 27979 . . . . 5 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (1st𝑊) ∈ Word dom (iEdg‘𝐺))
4 eqid 2740 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
54wlkp 27981 . . . . 5 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺))
63, 5jca 512 . . . 4 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)))
7 feq3 6581 . . . . . . 7 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶∅))
8 f00 6654 . . . . . . 7 ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶∅ ↔ ((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅))
97, 8bitrdi 287 . . . . . 6 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ ((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅)))
10 0z 12330 . . . . . . . . . . . . 13 0 ∈ ℤ
11 nn0z 12343 . . . . . . . . . . . . 13 ((♯‘(1st𝑊)) ∈ ℕ0 → (♯‘(1st𝑊)) ∈ ℤ)
12 fzn 13271 . . . . . . . . . . . . 13 ((0 ∈ ℤ ∧ (♯‘(1st𝑊)) ∈ ℤ) → ((♯‘(1st𝑊)) < 0 ↔ (0...(♯‘(1st𝑊))) = ∅))
1310, 11, 12sylancr 587 . . . . . . . . . . . 12 ((♯‘(1st𝑊)) ∈ ℕ0 → ((♯‘(1st𝑊)) < 0 ↔ (0...(♯‘(1st𝑊))) = ∅))
14 nn0nlt0 12259 . . . . . . . . . . . . 13 ((♯‘(1st𝑊)) ∈ ℕ0 → ¬ (♯‘(1st𝑊)) < 0)
1514pm2.21d 121 . . . . . . . . . . . 12 ((♯‘(1st𝑊)) ∈ ℕ0 → ((♯‘(1st𝑊)) < 0 → (1st𝑊) = ∅))
1613, 15sylbird 259 . . . . . . . . . . 11 ((♯‘(1st𝑊)) ∈ ℕ0 → ((0...(♯‘(1st𝑊))) = ∅ → (1st𝑊) = ∅))
1716com12 32 . . . . . . . . . 10 ((0...(♯‘(1st𝑊))) = ∅ → ((♯‘(1st𝑊)) ∈ ℕ0 → (1st𝑊) = ∅))
1817adantl 482 . . . . . . . . 9 (((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅) → ((♯‘(1st𝑊)) ∈ ℕ0 → (1st𝑊) = ∅))
19 lencl 14234 . . . . . . . . 9 ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → (♯‘(1st𝑊)) ∈ ℕ0)
2018, 19impel 506 . . . . . . . 8 ((((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → (1st𝑊) = ∅)
21 simpll 764 . . . . . . . 8 ((((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → (2nd𝑊) = ∅)
2220, 21jca 512 . . . . . . 7 ((((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
2322ex 413 . . . . . 6 (((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
249, 23syl6bi 252 . . . . 5 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))))
2524impcomd 412 . . . 4 ((Vtx‘𝐺) = ∅ → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
266, 25syl5 34 . . 3 ((Vtx‘𝐺) = ∅ → ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
271, 26syl5bi 241 . 2 ((Vtx‘𝐺) = ∅ → (𝑊 ∈ (Walks‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
2827imp 407 1 (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1542  wcel 2110  c0 4262   class class class wbr 5079  dom cdm 5590  wf 6428  cfv 6432  (class class class)co 7271  1st c1st 7822  2nd c2nd 7823  0cc0 10872   < clt 11010  0cn0 12233  cz 12319  ...cfz 13238  chash 14042  Word cword 14215  Vtxcvtx 27364  iEdgciedg 27365  Walkscwlks 27961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-er 8481  df-map 8600  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-card 9698  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12582  df-fz 13239  df-fzo 13382  df-hash 14043  df-word 14216  df-wlks 27964
This theorem is referenced by:  g0wlk0  28016
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