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Theorem wlkv0 27443
 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 27421 . . 3 (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))
2 eqid 2824 . . . . . 6 (iEdg‘𝐺) = (iEdg‘𝐺)
32wlkf 27407 . . . . 5 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (1st𝑊) ∈ Word dom (iEdg‘𝐺))
4 eqid 2824 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
54wlkp 27409 . . . . 5 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺))
63, 5jca 515 . . . 4 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)))
7 feq3 6486 . . . . . . 7 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶∅))
8 f00 6551 . . . . . . 7 ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶∅ ↔ ((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅))
97, 8syl6bb 290 . . . . . 6 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ ((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅)))
10 0z 11989 . . . . . . . . . . . . 13 0 ∈ ℤ
11 nn0z 12002 . . . . . . . . . . . . 13 ((♯‘(1st𝑊)) ∈ ℕ0 → (♯‘(1st𝑊)) ∈ ℤ)
12 fzn 12927 . . . . . . . . . . . . 13 ((0 ∈ ℤ ∧ (♯‘(1st𝑊)) ∈ ℤ) → ((♯‘(1st𝑊)) < 0 ↔ (0...(♯‘(1st𝑊))) = ∅))
1310, 11, 12sylancr 590 . . . . . . . . . . . 12 ((♯‘(1st𝑊)) ∈ ℕ0 → ((♯‘(1st𝑊)) < 0 ↔ (0...(♯‘(1st𝑊))) = ∅))
14 nn0nlt0 11920 . . . . . . . . . . . . 13 ((♯‘(1st𝑊)) ∈ ℕ0 → ¬ (♯‘(1st𝑊)) < 0)
1514pm2.21d 121 . . . . . . . . . . . 12 ((♯‘(1st𝑊)) ∈ ℕ0 → ((♯‘(1st𝑊)) < 0 → (1st𝑊) = ∅))
1613, 15sylbird 263 . . . . . . . . . . 11 ((♯‘(1st𝑊)) ∈ ℕ0 → ((0...(♯‘(1st𝑊))) = ∅ → (1st𝑊) = ∅))
1716com12 32 . . . . . . . . . 10 ((0...(♯‘(1st𝑊))) = ∅ → ((♯‘(1st𝑊)) ∈ ℕ0 → (1st𝑊) = ∅))
1817adantl 485 . . . . . . . . 9 (((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅) → ((♯‘(1st𝑊)) ∈ ℕ0 → (1st𝑊) = ∅))
19 lencl 13885 . . . . . . . . 9 ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → (♯‘(1st𝑊)) ∈ ℕ0)
2018, 19impel 509 . . . . . . . 8 ((((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → (1st𝑊) = ∅)
21 simpll 766 . . . . . . . 8 ((((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → (2nd𝑊) = ∅)
2220, 21jca 515 . . . . . . 7 ((((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
2322ex 416 . . . . . 6 (((2nd𝑊) = ∅ ∧ (0...(♯‘(1st𝑊))) = ∅) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
249, 23syl6bi 256 . . . . 5 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))))
2524impcomd 415 . . . 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 245 . 2 ((Vtx‘𝐺) = ∅ → (𝑊 ∈ (Walks‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
2827imp 410 1 (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∅c0 4276   class class class wbr 5052  dom cdm 5542  ⟶wf 6339  ‘cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  0cc0 10535   < clt 10673  ℕ0cn0 11894  ℤcz 11978  ...cfz 12894  ♯chash 13695  Word cword 13866  Vtxcvtx 26792  iEdgciedg 26793  Walkscwlks 27389 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-n0 11895  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-hash 13696  df-word 13867  df-wlks 27392 This theorem is referenced by:  g0wlk0  27444
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