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Mirrors > Home > MPE Home > Th. List > 0ewlk | Structured version Visualization version GIF version |
Description: The empty set (empty sequence of edges) is an s-walk of edges for all s. (Contributed by AV, 4-Jan-2021.) |
Ref | Expression |
---|---|
0ewlk | ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) → ∅ ∈ (𝐺 EdgWalks 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrd0 13627 | . . 3 ⊢ ∅ ∈ Word dom (iEdg‘𝐺) | |
2 | ral0 4299 | . . . 4 ⊢ ∀𝑘 ∈ ∅ 𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(∅‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(∅‘𝑘)))) | |
3 | hash0 13473 | . . . . . . 7 ⊢ (♯‘∅) = 0 | |
4 | 3 | oveq2i 6933 | . . . . . 6 ⊢ (1..^(♯‘∅)) = (1..^0) |
5 | 0le1 10898 | . . . . . . 7 ⊢ 0 ≤ 1 | |
6 | 1z 11759 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
7 | 0z 11739 | . . . . . . . . 9 ⊢ 0 ∈ ℤ | |
8 | 6, 7 | pm3.2i 464 | . . . . . . . 8 ⊢ (1 ∈ ℤ ∧ 0 ∈ ℤ) |
9 | fzon 12808 | . . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℤ) → (0 ≤ 1 ↔ (1..^0) = ∅)) | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 ⊢ (0 ≤ 1 ↔ (1..^0) = ∅) |
11 | 5, 10 | mpbi 222 | . . . . . 6 ⊢ (1..^0) = ∅ |
12 | 4, 11 | eqtri 2802 | . . . . 5 ⊢ (1..^(♯‘∅)) = ∅ |
13 | 12 | raleqi 3338 | . . . 4 ⊢ (∀𝑘 ∈ (1..^(♯‘∅))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(∅‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(∅‘𝑘)))) ↔ ∀𝑘 ∈ ∅ 𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(∅‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(∅‘𝑘))))) |
14 | 2, 13 | mpbir 223 | . . 3 ⊢ ∀𝑘 ∈ (1..^(♯‘∅))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(∅‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(∅‘𝑘)))) |
15 | 1, 14 | pm3.2i 464 | . 2 ⊢ (∅ ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘∅))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(∅‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(∅‘𝑘))))) |
16 | 0ex 5026 | . . 3 ⊢ ∅ ∈ V | |
17 | eqid 2778 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
18 | 17 | isewlk 26950 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ∧ ∅ ∈ V) → (∅ ∈ (𝐺 EdgWalks 𝑆) ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘∅))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(∅‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(∅‘𝑘))))))) |
19 | 16, 18 | mp3an3 1523 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) → (∅ ∈ (𝐺 EdgWalks 𝑆) ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘∅))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(∅‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(∅‘𝑘))))))) |
20 | 15, 19 | mpbiri 250 | 1 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) → ∅ ∈ (𝐺 EdgWalks 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 Vcvv 3398 ∩ cin 3791 ∅c0 4141 class class class wbr 4886 dom cdm 5355 ‘cfv 6135 (class class class)co 6922 0cc0 10272 1c1 10273 ≤ cle 10412 − cmin 10606 ℕ0*cxnn0 11714 ℤcz 11728 ..^cfzo 12784 ♯chash 13435 Word cword 13599 iEdgciedg 26345 EdgWalks cewlks 26943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-ewlks 26946 |
This theorem is referenced by: (None) |
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