wlkv0 - Metamath Proof Explorer

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Theorem wlkv0 26288

Description: If there is a walk in an empty graph, it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.)

**Assertion**
Ref	Expression
wlkv0	⊢ (𝑊 ∈ (∅ Walks 𝐸) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))

**Proof of Theorem wlkv0**
Step	Hyp	Ref	Expression
1		wlkcpr 26057	. 2 ⊢ (𝑊 ∈ (∅ Walks 𝐸) ↔ (1^st ‘𝑊)(∅ Walks 𝐸)(2^nd ‘𝑊))
2		2mwlk 26049	. . 3 ⊢ ((1^st ‘𝑊)(∅ Walks 𝐸)(2^nd ‘𝑊) → ((1^st ‘𝑊) ∈ Word dom 𝐸 ∧ (2^nd ‘𝑊):(0...(#‘(1^st ‘𝑊)))⟶∅))
3		f00 6000	. . . 4 ⊢ ((2^nd ‘𝑊):(0...(#‘(1^st ‘𝑊)))⟶∅ ↔ ((2^nd ‘𝑊) = ∅ ∧ (0...(#‘(1^st ‘𝑊))) = ∅))
4		lencl 13179	. . . . . . 7 ⊢ ((1^st ‘𝑊) ∈ Word dom 𝐸 → (#‘(1^st ‘𝑊)) ∈ ℕ₀)
5		0z 11265	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
6		nn0z 11277	. . . . . . . . . 10 ⊢ ((#‘(1^st ‘𝑊)) ∈ ℕ₀ → (#‘(1^st ‘𝑊)) ∈ ℤ)
7		fzn 12228	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ (#‘(1^st ‘𝑊)) ∈ ℤ) → ((#‘(1^st ‘𝑊)) < 0 ↔ (0...(#‘(1^st ‘𝑊))) = ∅))
8	5, 6, 7	sylancr 694	. . . . . . . . 9 ⊢ ((#‘(1^st ‘𝑊)) ∈ ℕ₀ → ((#‘(1^st ‘𝑊)) < 0 ↔ (0...(#‘(1^st ‘𝑊))) = ∅))
9		nn0nlt0 11196	. . . . . . . . . 10 ⊢ ((#‘(1^st ‘𝑊)) ∈ ℕ₀ → ¬ (#‘(1^st ‘𝑊)) < 0)
10	9	pm2.21d 117	. . . . . . . . 9 ⊢ ((#‘(1^st ‘𝑊)) ∈ ℕ₀ → ((#‘(1^st ‘𝑊)) < 0 → (1^st ‘𝑊) = ∅))
11	8, 10	sylbird 249	. . . . . . . 8 ⊢ ((#‘(1^st ‘𝑊)) ∈ ℕ₀ → ((0...(#‘(1^st ‘𝑊))) = ∅ → (1^st ‘𝑊) = ∅))
12	11	adantld 482	. . . . . . 7 ⊢ ((#‘(1^st ‘𝑊)) ∈ ℕ₀ → (((2^nd ‘𝑊) = ∅ ∧ (0...(#‘(1^st ‘𝑊))) = ∅) → (1^st ‘𝑊) = ∅))
13	4, 12	syl 17	. . . . . 6 ⊢ ((1^st ‘𝑊) ∈ Word dom 𝐸 → (((2^nd ‘𝑊) = ∅ ∧ (0...(#‘(1^st ‘𝑊))) = ∅) → (1^st ‘𝑊) = ∅))
14	13	imp 444	. . . . 5 ⊢ (((1^st ‘𝑊) ∈ Word dom 𝐸 ∧ ((2^nd ‘𝑊) = ∅ ∧ (0...(#‘(1^st ‘𝑊))) = ∅)) → (1^st ‘𝑊) = ∅)
15		simprl 790	. . . . 5 ⊢ (((1^st ‘𝑊) ∈ Word dom 𝐸 ∧ ((2^nd ‘𝑊) = ∅ ∧ (0...(#‘(1^st ‘𝑊))) = ∅)) → (2^nd ‘𝑊) = ∅)
16	14, 15	jca 553	. . . 4 ⊢ (((1^st ‘𝑊) ∈ Word dom 𝐸 ∧ ((2^nd ‘𝑊) = ∅ ∧ (0...(#‘(1^st ‘𝑊))) = ∅)) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))
17	3, 16	sylan2b 491	. . 3 ⊢ (((1^st ‘𝑊) ∈ Word dom 𝐸 ∧ (2^nd ‘𝑊):(0...(#‘(1^st ‘𝑊)))⟶∅) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))
18	2, 17	syl 17	. 2 ⊢ ((1^st ‘𝑊)(∅ Walks 𝐸)(2^nd ‘𝑊) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))
19	1, 18	sylbi 206	1 ⊢ (𝑊 ∈ (∅ Walks 𝐸) → ((1^st ‘𝑊) = ∅ ∧ (2^nd ‘𝑊) = ∅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 0cc0 9815 < clt 9953 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 #chash 12979 Word cword 13146 Walks cwalk 26026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-wlk 26036

This theorem is referenced by: wlk0 26289

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