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Theorem wlkv0 29908

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 29887	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
2		eqid 2765	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	2	wlkf 29873	. . . . 5 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺))
4		eqid 2765	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkp 29875	. . . . 5 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺))
6	3, 5	jca 520	. . . 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 6675	. . . . . . 7 ⊢ ((Vtx‘𝐺) = ∅ → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ↔ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶∅))
8		f00 6750	. . . . . . 7 ⊢ ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶∅ ↔ ((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅))
9	7, 8	bitrdi 290	. . . . . 6 ⊢ ((Vtx‘𝐺) = ∅ → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ↔ ((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅)))
10		0z 12593	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
11		nn0z 12606	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (♯‘(1^st ‘𝑊)) ∈ ℤ)
12		fzn 13559	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℤ ∧ (♯‘(1^st ‘𝑊)) ∈ ℤ) → ((♯‘(1^st ‘𝑊)) < 0 ↔ (0...(♯‘(1^st ‘𝑊))) = ∅))
13	10, 11, 12	sylancr 598	. . . . . . . . . . . 12 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → ((♯‘(1^st ‘𝑊)) < 0 ↔ (0...(♯‘(1^st ‘𝑊))) = ∅))
14		nn0nlt0 12521	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → ¬ (♯‘(1^st ‘𝑊)) < 0)
15	14	pm2.21d 122	. . . . . . . . . . . 12 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → ((♯‘(1^st ‘𝑊)) < 0 → (1^st ‘𝑊) = ∅))
16	13, 15	sylbird 263	. . . . . . . . . . 11 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → ((0...(♯‘(1^st ‘𝑊))) = ∅ → (1^st ‘𝑊) = ∅))
17	16	com12 33	. . . . . . . . . 10 ⊢ ((0...(♯‘(1^st ‘𝑊))) = ∅ → ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (1^st ‘𝑊) = ∅))
18	17	adantl 486	. . . . . . . . 9 ⊢ (((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) → ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (1^st ‘𝑊) = ∅))
19		lencl 14560	. . . . . . . . 9 ⊢ ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) → (♯‘(1^st ‘𝑊)) ∈ ℕ₀)
20	18, 19	impel 514	. . . . . . . 8 ⊢ ((((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) ∧ (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺)) → (1^st ‘𝑊) = ∅)
21		simpll 778	. . . . . . . 8 ⊢ ((((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) ∧ (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺)) → (2^nd ‘𝑊) = ∅)
22	20, 21	jca 520	. . . . . . 7 ⊢ ((((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) ∧ (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺)) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))
23	22	ex 417	. . . . . 6 ⊢ (((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) → ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅)))
24	9, 23	biimtrdi 256	. . . . 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 416	. . . 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 35	. . 3 ⊢ ((Vtx‘𝐺) = ∅ → ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅)))
27	1, 26	biimtrid 245	. 2 ⊢ ((Vtx‘𝐺) = ∅ → (𝑊 ∈ (Walks‘𝐺) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅)))
28	27	imp 411	1 ⊢ (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ∅c0 4288 class class class wbr 5105 dom cdm 5652 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 1^st c1st 7972 2^nd c2nd 7973 0cc0 11088 < clt 11231 ℕ₀cn0 12495 ℤcz 12582 ...cfz 13526 ♯chash 14357 Word cword 14540 Vtxcvtx 29255 iEdgciedg 29256 Walkscwlks 29855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1077 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-fzo 13674 df-hash 14358 df-word 14541 df-wlks 29858

This theorem is referenced by: g0wlk0 29909

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