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Mirrors > Home > MPE Home > Th. List > wlkreslem | Structured version Visualization version GIF version |
Description: Lemma for wlkres 27019. (Contributed by AV, 5-Mar-2021.) (Revised by AV, 30-Nov-2022.) |
Ref | Expression |
---|---|
wlkres.v | ⊢ 𝑉 = (Vtx‘𝐺) |
wlkres.i | ⊢ 𝐼 = (iEdg‘𝐺) |
wlkres.d | ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
wlkres.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
wlkres.s | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
Ref | Expression |
---|---|
wlkreslem | ⊢ (𝜑 → 𝑆 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . 2 ⊢ (𝑆 ∈ V → (𝜑 → 𝑆 ∈ V)) | |
2 | df-nel 3075 | . . 3 ⊢ (𝑆 ∉ V ↔ ¬ 𝑆 ∈ V) | |
3 | wlkres.d | . . . . 5 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
4 | df-br 4887 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (Walks‘𝐺)) | |
5 | ne0i 4148 | . . . . . . 7 ⊢ (〈𝐹, 𝑃〉 ∈ (Walks‘𝐺) → (Walks‘𝐺) ≠ ∅) | |
6 | wlkres.s | . . . . . . . . . . . 12 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
7 | wlkres.v | . . . . . . . . . . . 12 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 6, 7 | syl6eq 2829 | . . . . . . . . . . 11 ⊢ (𝜑 → (Vtx‘𝑆) = (Vtx‘𝐺)) |
9 | 8 | anim1ci 609 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 ∉ V) → (𝑆 ∉ V ∧ (Vtx‘𝑆) = (Vtx‘𝐺))) |
10 | wlk0prc 27001 | . . . . . . . . . 10 ⊢ ((𝑆 ∉ V ∧ (Vtx‘𝑆) = (Vtx‘𝐺)) → (Walks‘𝐺) = ∅) | |
11 | eqneqall 2979 | . . . . . . . . . 10 ⊢ ((Walks‘𝐺) = ∅ → ((Walks‘𝐺) ≠ ∅ → 𝑆 ∈ V)) | |
12 | 9, 10, 11 | 3syl 18 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 ∉ V) → ((Walks‘𝐺) ≠ ∅ → 𝑆 ∈ V)) |
13 | 12 | expcom 404 | . . . . . . . 8 ⊢ (𝑆 ∉ V → (𝜑 → ((Walks‘𝐺) ≠ ∅ → 𝑆 ∈ V))) |
14 | 13 | com13 88 | . . . . . . 7 ⊢ ((Walks‘𝐺) ≠ ∅ → (𝜑 → (𝑆 ∉ V → 𝑆 ∈ V))) |
15 | 5, 14 | syl 17 | . . . . . 6 ⊢ (〈𝐹, 𝑃〉 ∈ (Walks‘𝐺) → (𝜑 → (𝑆 ∉ V → 𝑆 ∈ V))) |
16 | 4, 15 | sylbi 209 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝜑 → (𝑆 ∉ V → 𝑆 ∈ V))) |
17 | 3, 16 | mpcom 38 | . . . 4 ⊢ (𝜑 → (𝑆 ∉ V → 𝑆 ∈ V)) |
18 | 17 | com12 32 | . . 3 ⊢ (𝑆 ∉ V → (𝜑 → 𝑆 ∈ V)) |
19 | 2, 18 | sylbir 227 | . 2 ⊢ (¬ 𝑆 ∈ V → (𝜑 → 𝑆 ∈ V)) |
20 | 1, 19 | pm2.61i 177 | 1 ⊢ (𝜑 → 𝑆 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∉ wnel 3074 Vcvv 3397 ∅c0 4140 〈cop 4403 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 0cc0 10272 ..^cfzo 12784 ♯chash 13435 Vtxcvtx 26344 iEdgciedg 26345 Walkscwlks 26944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ifp 1047 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-wlks 26947 |
This theorem is referenced by: wlkres 27019 |
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