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Mirrors > Home > MPE Home > Th. List > Mathboxes > lfuhgr2 | Structured version Visualization version GIF version |
Description: A hypergraph is loop-free if and only if every edge is not a loop. (Contributed by BTernaryTau, 15-Oct-2023.) |
Ref | Expression |
---|---|
lfuhgr.1 | ⊢ 𝑉 = (Vtx‘𝐺) |
lfuhgr.2 | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
lfuhgr2 | ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lfuhgr.1 | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | lfuhgr.2 | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | lfuhgr 32792 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
4 | uhgredgn0 27219 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) | |
5 | eldifsni 4703 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑥 ≠ ∅) | |
6 | 4, 5 | syl 17 | . . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 𝑥 ≠ ∅) |
7 | hashneq0 13931 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V → (0 < (♯‘𝑥) ↔ 𝑥 ≠ ∅)) | |
8 | 7 | elv 3414 | . . . . . . . . 9 ⊢ (0 < (♯‘𝑥) ↔ 𝑥 ≠ ∅) |
9 | 6, 8 | sylibr 237 | . . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 0 < (♯‘𝑥)) |
10 | 9 | gt0ne0d 11396 | . . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → (♯‘𝑥) ≠ 0) |
11 | hashxnn0 13905 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V → (♯‘𝑥) ∈ ℕ0*) | |
12 | 11 | elv 3414 | . . . . . . . . 9 ⊢ (♯‘𝑥) ∈ ℕ0* |
13 | xnn0n0n1ge2b 12723 | . . . . . . . . 9 ⊢ ((♯‘𝑥) ∈ ℕ0* → (((♯‘𝑥) ≠ 0 ∧ (♯‘𝑥) ≠ 1) ↔ 2 ≤ (♯‘𝑥))) | |
14 | 12, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (((♯‘𝑥) ≠ 0 ∧ (♯‘𝑥) ≠ 1) ↔ 2 ≤ (♯‘𝑥)) |
15 | 14 | biimpi 219 | . . . . . . 7 ⊢ (((♯‘𝑥) ≠ 0 ∧ (♯‘𝑥) ≠ 1) → 2 ≤ (♯‘𝑥)) |
16 | 10, 15 | stoic3 1784 | . . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ (♯‘𝑥) ≠ 1) → 2 ≤ (♯‘𝑥)) |
17 | 16 | 3exp 1121 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → (𝑥 ∈ (Edg‘𝐺) → ((♯‘𝑥) ≠ 1 → 2 ≤ (♯‘𝑥)))) |
18 | 17 | a2d 29 | . . . 4 ⊢ (𝐺 ∈ UHGraph → ((𝑥 ∈ (Edg‘𝐺) → (♯‘𝑥) ≠ 1) → (𝑥 ∈ (Edg‘𝐺) → 2 ≤ (♯‘𝑥)))) |
19 | 18 | ralimdv2 3099 | . . 3 ⊢ (𝐺 ∈ UHGraph → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1 → ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
20 | 1xr 10892 | . . . . 5 ⊢ 1 ∈ ℝ* | |
21 | hashxrcl 13924 | . . . . . 6 ⊢ (𝑥 ∈ V → (♯‘𝑥) ∈ ℝ*) | |
22 | 21 | elv 3414 | . . . . 5 ⊢ (♯‘𝑥) ∈ ℝ* |
23 | 1lt2 12001 | . . . . . 6 ⊢ 1 < 2 | |
24 | 2re 11904 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
25 | 24 | rexri 10891 | . . . . . . 7 ⊢ 2 ∈ ℝ* |
26 | xrltletr 12747 | . . . . . . 7 ⊢ ((1 ∈ ℝ* ∧ 2 ∈ ℝ* ∧ (♯‘𝑥) ∈ ℝ*) → ((1 < 2 ∧ 2 ≤ (♯‘𝑥)) → 1 < (♯‘𝑥))) | |
27 | 20, 25, 22, 26 | mp3an 1463 | . . . . . 6 ⊢ ((1 < 2 ∧ 2 ≤ (♯‘𝑥)) → 1 < (♯‘𝑥)) |
28 | 23, 27 | mpan 690 | . . . . 5 ⊢ (2 ≤ (♯‘𝑥) → 1 < (♯‘𝑥)) |
29 | xrltne 12753 | . . . . 5 ⊢ ((1 ∈ ℝ* ∧ (♯‘𝑥) ∈ ℝ* ∧ 1 < (♯‘𝑥)) → (♯‘𝑥) ≠ 1) | |
30 | 20, 22, 28, 29 | mp3an12i 1467 | . . . 4 ⊢ (2 ≤ (♯‘𝑥) → (♯‘𝑥) ≠ 1) |
31 | 30 | ralimi 3083 | . . 3 ⊢ (∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥) → ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1) |
32 | 19, 31 | impbid1 228 | . 2 ⊢ (𝐺 ∈ UHGraph → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1 ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
33 | 3, 32 | bitr4d 285 | 1 ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 {crab 3065 Vcvv 3408 ∖ cdif 3863 ∅c0 4237 𝒫 cpw 4513 {csn 4541 class class class wbr 5053 dom cdm 5551 ⟶wf 6376 ‘cfv 6380 0cc0 10729 1c1 10730 ℝ*cxr 10866 < clt 10867 ≤ cle 10868 2c2 11885 ℕ0*cxnn0 12162 ♯chash 13896 Vtxcvtx 27087 iEdgciedg 27088 Edgcedg 27138 UHGraphcuhgr 27147 |
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 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-n0 12091 df-xnn0 12163 df-z 12177 df-uz 12439 df-fz 13096 df-hash 13897 df-edg 27139 df-uhgr 27149 |
This theorem is referenced by: lfuhgr3 32794 |
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