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Theorem wlkv0 29684

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 29662	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
2		eqid 2735	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	2	wlkf 29647	. . . . 5 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺))
4		eqid 2735	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkp 29649	. . . . 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 6719	. . . . . . 7 ⊢ ((Vtx‘𝐺) = ∅ → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ↔ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶∅))
8		f00 6791	. . . . . . 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 12622	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
11		nn0z 12636	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (♯‘(1^st ‘𝑊)) ∈ ℤ)
12		fzn 13577	. . . . . . . . . . . . 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 12550	. . . . . . . . . . . . 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 14568	. . . . . . . . 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 767	. . . . . . . 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 253	. . . . 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 242	. 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∅c0 4339 class class class wbr 5148 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 0cc0 11153 < clt 11293 ℕ₀cn0 12524 ℤcz 12611 ...cfz 13544 ♯chash 14366 Word cword 14549 Vtxcvtx 29028 iEdgciedg 29029 Walkscwlks 29629

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-wlks 29632

This theorem is referenced by: g0wlk0 29685

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