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Theorem wlkv0 29965

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 29944	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
2		eqid 2761	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	2	wlkf 29930	. . . . 5 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺))
4		eqid 2761	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkp 29932	. . . . 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 6685	. . . . . . 7 ⊢ ((Vtx‘𝐺) = ∅ → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ↔ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶∅))
8		f00 6760	. . . . . . 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 12601	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
11		nn0z 12614	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (♯‘(1^st ‘𝑊)) ∈ ℤ)
12		fzn 13567	. . . . . . . . . . . . 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 12529	. . . . . . . . . . . . 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 14570	. . . . . . . . 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 1568 ∈ wcel 2141 ∅c0 4285 class class class wbr 5108 dom cdm 5661 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7983 2^nd c2nd 7984 0cc0 11099 < clt 11242 ℕ₀cn0 12503 ℤcz 12590 ...cfz 13534 ♯chash 14366 Word cword 14550 Vtxcvtx 29312 iEdgciedg 29313 Walkscwlks 29912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

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 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-n0 12504 df-z 12591 df-uz 12862 df-fz 13535 df-fzo 13683 df-hash 14367 df-word 14551 df-wlks 29915

This theorem is referenced by: g0wlk0 29966

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