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Theorem wlkv0 28018

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)) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))

**Proof of Theorem wlkv0**
Step	Hyp	Ref	Expression
1		wlkcpr 27996	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
2		eqid 2738	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	2	wlkf 27981	. . . . 5 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺))
4		eqid 2738	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkp 27983	. . . . 5 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺))
6	3, 5	jca 512	. . . 4 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)))
7		feq3 6583	. . . . . . 7 ⊢ ((Vtx‘𝐺) = ∅ → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ↔ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶∅))
8		f00 6656	. . . . . . 7 ⊢ ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶∅ ↔ ((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅))
9	7, 8	bitrdi 287	. . . . . 6 ⊢ ((Vtx‘𝐺) = ∅ → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ↔ ((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅)))
10		0z 12330	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
11		nn0z 12343	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (♯‘(1^st ‘𝑊)) ∈ ℤ)
12		fzn 13272	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℤ ∧ (♯‘(1^st ‘𝑊)) ∈ ℤ) → ((♯‘(1^st ‘𝑊)) < 0 ↔ (0...(♯‘(1^st ‘𝑊))) = ∅))
13	10, 11, 12	sylancr 587	. . . . . . . . . . . 12 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → ((♯‘(1^st ‘𝑊)) < 0 ↔ (0...(♯‘(1^st ‘𝑊))) = ∅))
14		nn0nlt0 12259	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → ¬ (♯‘(1^st ‘𝑊)) < 0)
15	14	pm2.21d 121	. . . . . . . . . . . 12 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → ((♯‘(1^st ‘𝑊)) < 0 → (1^st ‘𝑊) = ∅))
16	13, 15	sylbird 259	. . . . . . . . . . 11 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → ((0...(♯‘(1^st ‘𝑊))) = ∅ → (1^st ‘𝑊) = ∅))
17	16	com12 32	. . . . . . . . . 10 ⊢ ((0...(♯‘(1^st ‘𝑊))) = ∅ → ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (1^st ‘𝑊) = ∅))
18	17	adantl 482	. . . . . . . . 9 ⊢ (((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) → ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (1^st ‘𝑊) = ∅))
19		lencl 14236	. . . . . . . . 9 ⊢ ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) → (♯‘(1^st ‘𝑊)) ∈ ℕ₀)
20	18, 19	impel 506	. . . . . . . 8 ⊢ ((((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) ∧ (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺)) → (1^st ‘𝑊) = ∅)
21		simpll 764	. . . . . . . 8 ⊢ ((((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) ∧ (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺)) → (2^nd ‘𝑊) = ∅)
22	20, 21	jca 512	. . . . . . 7 ⊢ ((((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) ∧ (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺)) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))
23	22	ex 413	. . . . . 6 ⊢ (((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) → ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅)))
24	9, 23	syl6bi 252	. . . . 5 ⊢ ((Vtx‘𝐺) = ∅ → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) → ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))))
25	24	impcomd 412	. . . 4 ⊢ ((Vtx‘𝐺) = ∅ → (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅)))
26	6, 25	syl5 34	. . 3 ⊢ ((Vtx‘𝐺) = ∅ → ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅)))
27	1, 26	syl5bi 241	. 2 ⊢ ((Vtx‘𝐺) = ∅ → (𝑊 ∈ (Walks‘𝐺) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅)))
28	27	imp 407	1 ⊢ (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∅c0 4256 class class class wbr 5074 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 0cc0 10871 < clt 11009 ℕ₀cn0 12233 ℤcz 12319 ...cfz 13239 ♯chash 14044 Word cword 14217 Vtxcvtx 27366 iEdgciedg 27367 Walkscwlks 27963

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

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 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-wlks 27966

This theorem is referenced by: g0wlk0 28019

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