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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 2977 | . . . 4 ⊢ Ⅎ𝑥𝐼 | |
2 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | lfuhgrnloopv.e | . . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} | |
4 | nfrab1 3384 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} | |
5 | 3, 4 | nfcxfr 2975 | . . . 4 ⊢ Ⅎ𝑥𝐸 |
6 | 1, 2, 5 | nff 6510 | . . 3 ⊢ Ⅎ𝑥 𝐼:𝐴⟶𝐸 |
7 | hashsn01 13778 | . . . . . . 7 ⊢ ((♯‘{𝑈}) = 0 ∨ (♯‘{𝑈}) = 1) | |
8 | 2pos 11741 | . . . . . . . . . 10 ⊢ 0 < 2 | |
9 | 0re 10643 | . . . . . . . . . . 11 ⊢ 0 ∈ ℝ | |
10 | 2re 11712 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
11 | 9, 10 | ltnlei 10761 | . . . . . . . . . 10 ⊢ (0 < 2 ↔ ¬ 2 ≤ 0) |
12 | 8, 11 | mpbi 232 | . . . . . . . . 9 ⊢ ¬ 2 ≤ 0 |
13 | breq2 5070 | . . . . . . . . 9 ⊢ ((♯‘{𝑈}) = 0 → (2 ≤ (♯‘{𝑈}) ↔ 2 ≤ 0)) | |
14 | 12, 13 | mtbiri 329 | . . . . . . . 8 ⊢ ((♯‘{𝑈}) = 0 → ¬ 2 ≤ (♯‘{𝑈})) |
15 | 1lt2 11809 | . . . . . . . . . 10 ⊢ 1 < 2 | |
16 | 1re 10641 | . . . . . . . . . . 11 ⊢ 1 ∈ ℝ | |
17 | 16, 10 | ltnlei 10761 | . . . . . . . . . 10 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1) |
18 | 15, 17 | mpbi 232 | . . . . . . . . 9 ⊢ ¬ 2 ≤ 1 |
19 | breq2 5070 | . . . . . . . . 9 ⊢ ((♯‘{𝑈}) = 1 → (2 ≤ (♯‘{𝑈}) ↔ 2 ≤ 1)) | |
20 | 18, 19 | mtbiri 329 | . . . . . . . 8 ⊢ ((♯‘{𝑈}) = 1 → ¬ 2 ≤ (♯‘{𝑈})) |
21 | 14, 20 | jaoi 853 | . . . . . . 7 ⊢ (((♯‘{𝑈}) = 0 ∨ (♯‘{𝑈}) = 1) → ¬ 2 ≤ (♯‘{𝑈})) |
22 | 7, 21 | ax-mp 5 | . . . . . 6 ⊢ ¬ 2 ≤ (♯‘{𝑈}) |
23 | fveq2 6670 | . . . . . . 7 ⊢ ((𝐼‘𝑥) = {𝑈} → (♯‘(𝐼‘𝑥)) = (♯‘{𝑈})) | |
24 | 23 | breq2d 5078 | . . . . . 6 ⊢ ((𝐼‘𝑥) = {𝑈} → (2 ≤ (♯‘(𝐼‘𝑥)) ↔ 2 ≤ (♯‘{𝑈}))) |
25 | 22, 24 | mtbiri 329 | . . . . 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 26909 | . . . . 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 415 | . . 3 ⊢ (𝐼:𝐴⟶𝐸 → (𝑥 ∈ 𝐴 → ¬ (𝐼‘𝑥) = {𝑈})) |
31 | 6, 30 | ralrimi 3216 | . 2 ⊢ (𝐼:𝐴⟶𝐸 → ∀𝑥 ∈ 𝐴 ¬ (𝐼‘𝑥) = {𝑈}) |
32 | rabeq0 4338 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐼‘𝑥) = {𝑈}) | |
33 | 31, 32 | sylibr 236 | 1 ⊢ (𝐼:𝐴⟶𝐸 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 ∅c0 4291 𝒫 cpw 4539 {csn 4567 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 0cc0 10537 1c1 10538 < clt 10675 ≤ cle 10676 2c2 11693 ♯chash 13691 iEdgciedg 26782 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 |
This theorem is referenced by: vtxdlfgrval 27267 |
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