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Mirrors > Home > MPE Home > Th. List > lfgrnloop | Structured version Visualization version GIF version |
Description: A loop-free graph has no loops. (Contributed by AV, 23-Feb-2021.) |
Ref | Expression |
---|---|
lfuhgrnloopv.i | ⊢ 𝐼 = (iEdg‘𝐺) |
lfuhgrnloopv.a | ⊢ 𝐴 = dom 𝐼 |
lfuhgrnloopv.e | ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} |
Ref | Expression |
---|---|
lfgrnloop | ⊢ (𝐼:𝐴⟶𝐸 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2933 | . . . 4 ⊢ Ⅎ𝑥𝐼 | |
2 | nfcv 2933 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | lfuhgrnloopv.e | . . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} | |
4 | nfrab1 3325 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} | |
5 | 3, 4 | nfcxfr 2931 | . . . 4 ⊢ Ⅎ𝑥𝐸 |
6 | 1, 2, 5 | nff 6340 | . . 3 ⊢ Ⅎ𝑥 𝐼:𝐴⟶𝐸 |
7 | hashsn01 13590 | . . . . . . 7 ⊢ ((♯‘{𝑈}) = 0 ∨ (♯‘{𝑈}) = 1) | |
8 | 2pos 11550 | . . . . . . . . . 10 ⊢ 0 < 2 | |
9 | 0re 10441 | . . . . . . . . . . 11 ⊢ 0 ∈ ℝ | |
10 | 2re 11514 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
11 | 9, 10 | ltnlei 10561 | . . . . . . . . . 10 ⊢ (0 < 2 ↔ ¬ 2 ≤ 0) |
12 | 8, 11 | mpbi 222 | . . . . . . . . 9 ⊢ ¬ 2 ≤ 0 |
13 | breq2 4933 | . . . . . . . . 9 ⊢ ((♯‘{𝑈}) = 0 → (2 ≤ (♯‘{𝑈}) ↔ 2 ≤ 0)) | |
14 | 12, 13 | mtbiri 319 | . . . . . . . 8 ⊢ ((♯‘{𝑈}) = 0 → ¬ 2 ≤ (♯‘{𝑈})) |
15 | 1lt2 11618 | . . . . . . . . . 10 ⊢ 1 < 2 | |
16 | 1re 10439 | . . . . . . . . . . 11 ⊢ 1 ∈ ℝ | |
17 | 16, 10 | ltnlei 10561 | . . . . . . . . . 10 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1) |
18 | 15, 17 | mpbi 222 | . . . . . . . . 9 ⊢ ¬ 2 ≤ 1 |
19 | breq2 4933 | . . . . . . . . 9 ⊢ ((♯‘{𝑈}) = 1 → (2 ≤ (♯‘{𝑈}) ↔ 2 ≤ 1)) | |
20 | 18, 19 | mtbiri 319 | . . . . . . . 8 ⊢ ((♯‘{𝑈}) = 1 → ¬ 2 ≤ (♯‘{𝑈})) |
21 | 14, 20 | jaoi 843 | . . . . . . 7 ⊢ (((♯‘{𝑈}) = 0 ∨ (♯‘{𝑈}) = 1) → ¬ 2 ≤ (♯‘{𝑈})) |
22 | 7, 21 | ax-mp 5 | . . . . . 6 ⊢ ¬ 2 ≤ (♯‘{𝑈}) |
23 | fveq2 6499 | . . . . . . 7 ⊢ ((𝐼‘𝑥) = {𝑈} → (♯‘(𝐼‘𝑥)) = (♯‘{𝑈})) | |
24 | 23 | breq2d 4941 | . . . . . 6 ⊢ ((𝐼‘𝑥) = {𝑈} → (2 ≤ (♯‘(𝐼‘𝑥)) ↔ 2 ≤ (♯‘{𝑈}))) |
25 | 22, 24 | mtbiri 319 | . . . . 5 ⊢ ((𝐼‘𝑥) = {𝑈} → ¬ 2 ≤ (♯‘(𝐼‘𝑥))) |
26 | lfuhgrnloopv.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
27 | lfuhgrnloopv.a | . . . . . 6 ⊢ 𝐴 = dom 𝐼 | |
28 | 26, 27, 3 | lfgredgge2 26612 | . . . . 5 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑥 ∈ 𝐴) → 2 ≤ (♯‘(𝐼‘𝑥))) |
29 | 25, 28 | nsyl3 136 | . . . 4 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑥 ∈ 𝐴) → ¬ (𝐼‘𝑥) = {𝑈}) |
30 | 29 | ex 405 | . . 3 ⊢ (𝐼:𝐴⟶𝐸 → (𝑥 ∈ 𝐴 → ¬ (𝐼‘𝑥) = {𝑈})) |
31 | 6, 30 | ralrimi 3167 | . 2 ⊢ (𝐼:𝐴⟶𝐸 → ∀𝑥 ∈ 𝐴 ¬ (𝐼‘𝑥) = {𝑈}) |
32 | rabeq0 4225 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐼‘𝑥) = {𝑈}) | |
33 | 31, 32 | sylibr 226 | 1 ⊢ (𝐼:𝐴⟶𝐸 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2050 ∀wral 3089 {crab 3093 ∅c0 4179 𝒫 cpw 4422 {csn 4441 class class class wbr 4929 dom cdm 5407 ⟶wf 6184 ‘cfv 6188 0cc0 10335 1c1 10336 < clt 10474 ≤ cle 10475 2c2 11495 ♯chash 13505 iEdgciedg 26485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-hash 13506 |
This theorem is referenced by: vtxdlfgrval 26970 |
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