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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 32364 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
4 | uhgredgn0 26913 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) | |
5 | eldifsni 4722 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑥 ≠ ∅) | |
6 | 4, 5 | syl 17 | . . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 𝑥 ≠ ∅) |
7 | hashneq0 13726 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V → (0 < (♯‘𝑥) ↔ 𝑥 ≠ ∅)) | |
8 | 7 | elv 3499 | . . . . . . . . 9 ⊢ (0 < (♯‘𝑥) ↔ 𝑥 ≠ ∅) |
9 | 6, 8 | sylibr 236 | . . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 0 < (♯‘𝑥)) |
10 | 9 | gt0ne0d 11204 | . . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → (♯‘𝑥) ≠ 0) |
11 | hashxnn0 13700 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V → (♯‘𝑥) ∈ ℕ0*) | |
12 | 11 | elv 3499 | . . . . . . . . 9 ⊢ (♯‘𝑥) ∈ ℕ0* |
13 | xnn0n0n1ge2b 12527 | . . . . . . . . 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 218 | . . . . . . 7 ⊢ (((♯‘𝑥) ≠ 0 ∧ (♯‘𝑥) ≠ 1) → 2 ≤ (♯‘𝑥)) |
16 | 10, 15 | stoic3 1777 | . . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ (♯‘𝑥) ≠ 1) → 2 ≤ (♯‘𝑥)) |
17 | 16 | 3exp 1115 | . . . . 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 3176 | . . 3 ⊢ (𝐺 ∈ UHGraph → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1 → ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
20 | 1xr 10700 | . . . . 5 ⊢ 1 ∈ ℝ* | |
21 | hashxrcl 13719 | . . . . . 6 ⊢ (𝑥 ∈ V → (♯‘𝑥) ∈ ℝ*) | |
22 | 21 | elv 3499 | . . . . 5 ⊢ (♯‘𝑥) ∈ ℝ* |
23 | 1lt2 11809 | . . . . . 6 ⊢ 1 < 2 | |
24 | 2re 11712 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
25 | 24 | rexri 10699 | . . . . . . 7 ⊢ 2 ∈ ℝ* |
26 | xrltletr 12551 | . . . . . . 7 ⊢ ((1 ∈ ℝ* ∧ 2 ∈ ℝ* ∧ (♯‘𝑥) ∈ ℝ*) → ((1 < 2 ∧ 2 ≤ (♯‘𝑥)) → 1 < (♯‘𝑥))) | |
27 | 20, 25, 22, 26 | mp3an 1457 | . . . . . 6 ⊢ ((1 < 2 ∧ 2 ≤ (♯‘𝑥)) → 1 < (♯‘𝑥)) |
28 | 23, 27 | mpan 688 | . . . . 5 ⊢ (2 ≤ (♯‘𝑥) → 1 < (♯‘𝑥)) |
29 | xrltne 12557 | . . . . 5 ⊢ ((1 ∈ ℝ* ∧ (♯‘𝑥) ∈ ℝ* ∧ 1 < (♯‘𝑥)) → (♯‘𝑥) ≠ 1) | |
30 | 20, 22, 28, 29 | mp3an12i 1461 | . . . 4 ⊢ (2 ≤ (♯‘𝑥) → (♯‘𝑥) ≠ 1) |
31 | 30 | ralimi 3160 | . . 3 ⊢ (∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥) → ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1) |
32 | 19, 31 | impbid1 227 | . 2 ⊢ (𝐺 ∈ UHGraph → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1 ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
33 | 3, 32 | bitr4d 284 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 {crab 3142 Vcvv 3494 ∖ cdif 3933 ∅c0 4291 𝒫 cpw 4539 {csn 4567 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 0cc0 10537 1c1 10538 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 2c2 11693 ℕ0*cxnn0 11968 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 Edgcedg 26832 UHGraphcuhgr 26841 |
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-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 df-edg 26833 df-uhgr 26843 |
This theorem is referenced by: lfuhgr3 32366 |
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