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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 35087 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
4 | uhgredgn0 29165 | . . . . . . . . . 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 14415 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V → (0 < (♯‘𝑥) ↔ 𝑥 ≠ ∅)) | |
8 | 7 | elv 3493 | . . . . . . . . 9 ⊢ (0 < (♯‘𝑥) ↔ 𝑥 ≠ ∅) |
9 | 6, 8 | sylibr 234 | . . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 0 < (♯‘𝑥)) |
10 | 9 | gt0ne0d 11856 | . . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → (♯‘𝑥) ≠ 0) |
11 | hashxnn0 14390 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V → (♯‘𝑥) ∈ ℕ0*) | |
12 | 11 | elv 3493 | . . . . . . . . 9 ⊢ (♯‘𝑥) ∈ ℕ0* |
13 | xnn0n0n1ge2b 13196 | . . . . . . . . 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 3169 | . . 3 ⊢ (𝐺 ∈ UHGraph → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1 → ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
20 | 1xr 11351 | . . . . 5 ⊢ 1 ∈ ℝ* | |
21 | hashxrcl 14408 | . . . . . 6 ⊢ (𝑥 ∈ V → (♯‘𝑥) ∈ ℝ*) | |
22 | 21 | elv 3493 | . . . . 5 ⊢ (♯‘𝑥) ∈ ℝ* |
23 | 1lt2 12466 | . . . . . 6 ⊢ 1 < 2 | |
24 | 2re 12369 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
25 | 24 | rexri 11350 | . . . . . . 7 ⊢ 2 ∈ ℝ* |
26 | xrltletr 13221 | . . . . . . 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 13227 | . . . . 5 ⊢ ((1 ∈ ℝ* ∧ (♯‘𝑥) ∈ ℝ* ∧ 1 < (♯‘𝑥)) → (♯‘𝑥) ≠ 1) | |
30 | 20, 22, 28, 29 | mp3an12i 1465 | . . . 4 ⊢ (2 ≤ (♯‘𝑥) → (♯‘𝑥) ≠ 1) |
31 | 30 | ralimi 3089 | . . 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 2108 ≠ wne 2946 ∀wral 3067 {crab 3443 Vcvv 3488 ∖ cdif 3973 ∅c0 4352 𝒫 cpw 4622 {csn 4648 class class class wbr 5166 dom cdm 5700 ⟶wf 6571 ‘cfv 6575 0cc0 11186 1c1 11187 ℝ*cxr 11325 < clt 11326 ≤ cle 11327 2c2 12350 ℕ0*cxnn0 12627 ♯chash 14381 Vtxcvtx 29033 iEdgciedg 29034 Edgcedg 29084 UHGraphcuhgr 29093 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-n0 12556 df-xnn0 12628 df-z 12642 df-uz 12906 df-fz 13570 df-hash 14382 df-edg 29085 df-uhgr 29095 |
This theorem is referenced by: lfuhgr3 35089 |
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