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Theorem wlkv0 29459

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 29437	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
2		eqid 2728	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	2	wlkf 29422	. . . . 5 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺))
4		eqid 2728	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkp 29424	. . . . 5 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺))
6	3, 5	jca 511	. . . 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 6700	. . . . . . 7 ⊢ ((Vtx‘𝐺) = ∅ → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ↔ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶∅))
8		f00 6774	. . . . . . 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 12594	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
11		nn0z 12608	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (♯‘(1^st ‘𝑊)) ∈ ℤ)
12		fzn 13544	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℤ ∧ (♯‘(1^st ‘𝑊)) ∈ ℤ) → ((♯‘(1^st ‘𝑊)) < 0 ↔ (0...(♯‘(1^st ‘𝑊))) = ∅))
13	10, 11, 12	sylancr 586	. . . . . . . . . . . 12 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → ((♯‘(1^st ‘𝑊)) < 0 ↔ (0...(♯‘(1^st ‘𝑊))) = ∅))
14		nn0nlt0 12523	. . . . . . . . . . . . 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 260	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ (((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) → ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (1^st ‘𝑊) = ∅))
19		lencl 14510	. . . . . . . . 9 ⊢ ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) → (♯‘(1^st ‘𝑊)) ∈ ℕ₀)
20	18, 19	impel 505	. . . . . . . 8 ⊢ ((((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) ∧ (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺)) → (1^st ‘𝑊) = ∅)
21		simpll 766	. . . . . . . 8 ⊢ ((((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) ∧ (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺)) → (2^nd ‘𝑊) = ∅)
22	20, 21	jca 511	. . . . . . 7 ⊢ ((((2^nd ‘𝑊) = ∅ ∧ (0...(♯‘(1^st ‘𝑊))) = ∅) ∧ (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺)) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))
23	22	ex 412	. . . . . 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 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 411	. . . 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	biimtrid 241	. 2 ⊢ ((Vtx‘𝐺) = ∅ → (𝑊 ∈ (Walks‘𝐺) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅)))
28	27	imp 406	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 395 = wceq 1534 ∈ wcel 2099 ∅c0 4319 class class class wbr 5143 dom cdm 5673 ⟶wf 6539 ‘cfv 6543 (class class class)co 7415 1^st c1st 7986 2^nd c2nd 7987 0cc0 11133 < clt 11273 ℕ₀cn0 12497 ℤcz 12583 ...cfz 13511 ♯chash 14316 Word cword 14491 Vtxcvtx 28803 iEdgciedg 28804 Walkscwlks 29404

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-wlks 29407

This theorem is referenced by: g0wlk0 29460

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