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Mirrors > Home > MPE Home > Th. List > clwwlknfiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of clwwlknfi 27826 as of 4-May-2023. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
clwwlknfiOLD | ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 11907 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | clwwlkn 27807 | . . . . 5 ⊢ (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} | |
3 | wrdnfiOLD 13903 | . . . . . 6 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ Fin) | |
4 | clwwlksswrd 27768 | . . . . . . 7 ⊢ (ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) | |
5 | rabss2 4057 | . . . . . . 7 ⊢ ((ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ⊆ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁}) | |
6 | 4, 5 | mp1i 13 | . . . . . 6 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ⊆ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁}) |
7 | 3, 6 | ssfid 8744 | . . . . 5 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ Fin) |
8 | 2, 7 | eqeltrid 2920 | . . . 4 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
9 | 8 | expcom 416 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
10 | 1, 9 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
11 | df-nel 3127 | . . . . . . 7 ⊢ (𝑁 ∉ ℕ ↔ ¬ 𝑁 ∈ ℕ) | |
12 | 11 | biimpri 230 | . . . . . 6 ⊢ (¬ 𝑁 ∈ ℕ → 𝑁 ∉ ℕ) |
13 | 12 | olcd 870 | . . . . 5 ⊢ (¬ 𝑁 ∈ ℕ → (𝐺 ∉ V ∨ 𝑁 ∉ ℕ)) |
14 | clwwlkneq0 27810 | . . . . 5 ⊢ ((𝐺 ∉ V ∨ 𝑁 ∉ ℕ) → (𝑁 ClWWalksN 𝐺) = ∅) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (¬ 𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = ∅) |
16 | 0fin 8749 | . . . 4 ⊢ ∅ ∈ Fin | |
17 | 15, 16 | eqeltrdi 2924 | . . 3 ⊢ (¬ 𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
18 | 17 | a1d 25 | . 2 ⊢ (¬ 𝑁 ∈ ℕ → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
19 | 10, 18 | pm2.61i 184 | 1 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ∉ wnel 3126 {crab 3145 Vcvv 3497 ⊆ wss 3939 ∅c0 4294 ‘cfv 6358 (class class class)co 7159 Fincfn 8512 ℕcn 11641 ℕ0cn0 11900 ♯chash 13693 Word cword 13864 Vtxcvtx 26784 ClWWalkscclwwlk 27762 ClWWalksN cclwwlkn 27805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-word 13865 df-clwwlk 27763 df-clwwlkn 27806 |
This theorem is referenced by: (None) |
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