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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 35077 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
4 | uhgredgn0 29154 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) | |
5 | eldifsni 4815 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑥 ≠ ∅) | |
6 | 4, 5 | syl 17 | . . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 𝑥 ≠ ∅) |
7 | hashneq0 14409 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V → (0 < (♯‘𝑥) ↔ 𝑥 ≠ ∅)) | |
8 | 7 | elv 3488 | . . . . . . . . 9 ⊢ (0 < (♯‘𝑥) ↔ 𝑥 ≠ ∅) |
9 | 6, 8 | sylibr 234 | . . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 0 < (♯‘𝑥)) |
10 | 9 | gt0ne0d 11850 | . . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → (♯‘𝑥) ≠ 0) |
11 | hashxnn0 14384 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V → (♯‘𝑥) ∈ ℕ0*) | |
12 | 11 | elv 3488 | . . . . . . . . 9 ⊢ (♯‘𝑥) ∈ ℕ0* |
13 | xnn0n0n1ge2b 13190 | . . . . . . . . 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 216 | . . . . . . 7 ⊢ (((♯‘𝑥) ≠ 0 ∧ (♯‘𝑥) ≠ 1) → 2 ≤ (♯‘𝑥)) |
16 | 10, 15 | stoic3 1774 | . . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ (♯‘𝑥) ≠ 1) → 2 ≤ (♯‘𝑥)) |
17 | 16 | 3exp 1119 | . . . . 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 3165 | . . 3 ⊢ (𝐺 ∈ UHGraph → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1 → ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
20 | 1xr 11345 | . . . . 5 ⊢ 1 ∈ ℝ* | |
21 | hashxrcl 14402 | . . . . . 6 ⊢ (𝑥 ∈ V → (♯‘𝑥) ∈ ℝ*) | |
22 | 21 | elv 3488 | . . . . 5 ⊢ (♯‘𝑥) ∈ ℝ* |
23 | 1lt2 12460 | . . . . . 6 ⊢ 1 < 2 | |
24 | 2re 12363 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
25 | 24 | rexri 11344 | . . . . . . 7 ⊢ 2 ∈ ℝ* |
26 | xrltletr 13215 | . . . . . . 7 ⊢ ((1 ∈ ℝ* ∧ 2 ∈ ℝ* ∧ (♯‘𝑥) ∈ ℝ*) → ((1 < 2 ∧ 2 ≤ (♯‘𝑥)) → 1 < (♯‘𝑥))) | |
27 | 20, 25, 22, 26 | mp3an 1461 | . . . . . 6 ⊢ ((1 < 2 ∧ 2 ≤ (♯‘𝑥)) → 1 < (♯‘𝑥)) |
28 | 23, 27 | mpan 689 | . . . . 5 ⊢ (2 ≤ (♯‘𝑥) → 1 < (♯‘𝑥)) |
29 | xrltne 13221 | . . . . 5 ⊢ ((1 ∈ ℝ* ∧ (♯‘𝑥) ∈ ℝ* ∧ 1 < (♯‘𝑥)) → (♯‘𝑥) ≠ 1) | |
30 | 20, 22, 28, 29 | mp3an12i 1465 | . . . 4 ⊢ (2 ≤ (♯‘𝑥) → (♯‘𝑥) ≠ 1) |
31 | 30 | ralimi 3085 | . . 3 ⊢ (∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥) → ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1) |
32 | 19, 31 | impbid1 225 | . 2 ⊢ (𝐺 ∈ UHGraph → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1 ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
33 | 3, 32 | bitr4d 282 | 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 ∀wral 3063 {crab 3438 Vcvv 3482 ∖ cdif 3967 ∅c0 4347 𝒫 cpw 4622 {csn 4648 class class class wbr 5169 dom cdm 5699 ⟶wf 6568 ‘cfv 6572 0cc0 11180 1c1 11181 ℝ*cxr 11319 < clt 11320 ≤ cle 11321 2c2 12344 ℕ0*cxnn0 12621 ♯chash 14375 Vtxcvtx 29022 iEdgciedg 29023 Edgcedg 29073 UHGraphcuhgr 29082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-n0 12550 df-xnn0 12622 df-z 12636 df-uz 12900 df-fz 13564 df-hash 14376 df-edg 29074 df-uhgr 29084 |
This theorem is referenced by: lfuhgr3 35079 |
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