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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 2900 | . . . 4 ⊢ Ⅎ𝑥𝐼 | |
2 | nfcv 2900 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | lfuhgrnloopv.e | . . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} | |
4 | nfrab1 3289 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} | |
5 | 3, 4 | nfcxfr 2898 | . . . 4 ⊢ Ⅎ𝑥𝐸 |
6 | 1, 2, 5 | nff 6530 | . . 3 ⊢ Ⅎ𝑥 𝐼:𝐴⟶𝐸 |
7 | hashsn01 13966 | . . . . . . 7 ⊢ ((♯‘{𝑈}) = 0 ∨ (♯‘{𝑈}) = 1) | |
8 | 2pos 11916 | . . . . . . . . . 10 ⊢ 0 < 2 | |
9 | 0re 10818 | . . . . . . . . . . 11 ⊢ 0 ∈ ℝ | |
10 | 2re 11887 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
11 | 9, 10 | ltnlei 10936 | . . . . . . . . . 10 ⊢ (0 < 2 ↔ ¬ 2 ≤ 0) |
12 | 8, 11 | mpbi 233 | . . . . . . . . 9 ⊢ ¬ 2 ≤ 0 |
13 | breq2 5047 | . . . . . . . . 9 ⊢ ((♯‘{𝑈}) = 0 → (2 ≤ (♯‘{𝑈}) ↔ 2 ≤ 0)) | |
14 | 12, 13 | mtbiri 330 | . . . . . . . 8 ⊢ ((♯‘{𝑈}) = 0 → ¬ 2 ≤ (♯‘{𝑈})) |
15 | 1lt2 11984 | . . . . . . . . . 10 ⊢ 1 < 2 | |
16 | 1re 10816 | . . . . . . . . . . 11 ⊢ 1 ∈ ℝ | |
17 | 16, 10 | ltnlei 10936 | . . . . . . . . . 10 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1) |
18 | 15, 17 | mpbi 233 | . . . . . . . . 9 ⊢ ¬ 2 ≤ 1 |
19 | breq2 5047 | . . . . . . . . 9 ⊢ ((♯‘{𝑈}) = 1 → (2 ≤ (♯‘{𝑈}) ↔ 2 ≤ 1)) | |
20 | 18, 19 | mtbiri 330 | . . . . . . . 8 ⊢ ((♯‘{𝑈}) = 1 → ¬ 2 ≤ (♯‘{𝑈})) |
21 | 14, 20 | jaoi 857 | . . . . . . 7 ⊢ (((♯‘{𝑈}) = 0 ∨ (♯‘{𝑈}) = 1) → ¬ 2 ≤ (♯‘{𝑈})) |
22 | 7, 21 | ax-mp 5 | . . . . . 6 ⊢ ¬ 2 ≤ (♯‘{𝑈}) |
23 | fveq2 6706 | . . . . . . 7 ⊢ ((𝐼‘𝑥) = {𝑈} → (♯‘(𝐼‘𝑥)) = (♯‘{𝑈})) | |
24 | 23 | breq2d 5055 | . . . . . 6 ⊢ ((𝐼‘𝑥) = {𝑈} → (2 ≤ (♯‘(𝐼‘𝑥)) ↔ 2 ≤ (♯‘{𝑈}))) |
25 | 22, 24 | mtbiri 330 | . . . . 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 27187 | . . . . 5 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑥 ∈ 𝐴) → 2 ≤ (♯‘(𝐼‘𝑥))) |
29 | 25, 28 | nsyl3 140 | . . . 4 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑥 ∈ 𝐴) → ¬ (𝐼‘𝑥) = {𝑈}) |
30 | 29 | ex 416 | . . 3 ⊢ (𝐼:𝐴⟶𝐸 → (𝑥 ∈ 𝐴 → ¬ (𝐼‘𝑥) = {𝑈})) |
31 | 6, 30 | ralrimi 3130 | . 2 ⊢ (𝐼:𝐴⟶𝐸 → ∀𝑥 ∈ 𝐴 ¬ (𝐼‘𝑥) = {𝑈}) |
32 | rabeq0 4289 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐼‘𝑥) = {𝑈}) | |
33 | 31, 32 | sylibr 237 | 1 ⊢ (𝐼:𝐴⟶𝐸 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ∀wral 3054 {crab 3058 ∅c0 4227 𝒫 cpw 4503 {csn 4531 class class class wbr 5043 dom cdm 5540 ⟶wf 6365 ‘cfv 6369 0cc0 10712 1c1 10713 < clt 10850 ≤ cle 10851 2c2 11868 ♯chash 13879 iEdgciedg 27060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-hash 13880 |
This theorem is referenced by: vtxdlfgrval 27545 |
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