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Mirrors > Home > MPE Home > Th. List > wlkreslem | Structured version Visualization version GIF version |
Description: Lemma for wlkres 27454. (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 3126 | . . 3 ⊢ (𝑆 ∉ V ↔ ¬ 𝑆 ∈ V) | |
3 | wlkres.d | . . . . 5 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
4 | df-br 5069 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (Walks‘𝐺)) | |
5 | ne0i 4302 | . . . . . . 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 2874 | . . . . . . . . . . 11 ⊢ (𝜑 → (Vtx‘𝑆) = (Vtx‘𝐺)) |
9 | 8 | anim1ci 617 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 ∉ V) → (𝑆 ∉ V ∧ (Vtx‘𝑆) = (Vtx‘𝐺))) |
10 | wlk0prc 27437 | . . . . . . . . . 10 ⊢ ((𝑆 ∉ V ∧ (Vtx‘𝑆) = (Vtx‘𝐺)) → (Walks‘𝐺) = ∅) | |
11 | eqneqall 3029 | . . . . . . . . . 10 ⊢ ((Walks‘𝐺) = ∅ → ((Walks‘𝐺) ≠ ∅ → 𝑆 ∈ V)) | |
12 | 9, 10, 11 | 3syl 18 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 ∉ V) → ((Walks‘𝐺) ≠ ∅ → 𝑆 ∈ V)) |
13 | 12 | expcom 416 | . . . . . . . 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 219 | . . . . 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 237 | . 2 ⊢ (¬ 𝑆 ∈ V → (𝜑 → 𝑆 ∈ V)) |
20 | 1, 19 | pm2.61i 184 | 1 ⊢ (𝜑 → 𝑆 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∉ wnel 3125 Vcvv 3496 ∅c0 4293 〈cop 4575 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 0cc0 10539 ..^cfzo 13036 ♯chash 13693 Vtxcvtx 26783 iEdgciedg 26784 Walkscwlks 27380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-wlks 27383 |
This theorem is referenced by: wlkres 27454 |
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