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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 33768 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
4 | uhgredgn0 28121 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) | |
5 | eldifsni 4751 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑥 ≠ ∅) | |
6 | 4, 5 | syl 17 | . . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 𝑥 ≠ ∅) |
7 | hashneq0 14270 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V → (0 < (♯‘𝑥) ↔ 𝑥 ≠ ∅)) | |
8 | 7 | elv 3450 | . . . . . . . . 9 ⊢ (0 < (♯‘𝑥) ↔ 𝑥 ≠ ∅) |
9 | 6, 8 | sylibr 233 | . . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 0 < (♯‘𝑥)) |
10 | 9 | gt0ne0d 11724 | . . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → (♯‘𝑥) ≠ 0) |
11 | hashxnn0 14245 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V → (♯‘𝑥) ∈ ℕ0*) | |
12 | 11 | elv 3450 | . . . . . . . . 9 ⊢ (♯‘𝑥) ∈ ℕ0* |
13 | xnn0n0n1ge2b 13057 | . . . . . . . . 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 215 | . . . . . . 7 ⊢ (((♯‘𝑥) ≠ 0 ∧ (♯‘𝑥) ≠ 1) → 2 ≤ (♯‘𝑥)) |
16 | 10, 15 | stoic3 1779 | . . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ (♯‘𝑥) ≠ 1) → 2 ≤ (♯‘𝑥)) |
17 | 16 | 3exp 1120 | . . . . 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 3157 | . . 3 ⊢ (𝐺 ∈ UHGraph → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1 → ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥))) |
20 | 1xr 11219 | . . . . 5 ⊢ 1 ∈ ℝ* | |
21 | hashxrcl 14263 | . . . . . 6 ⊢ (𝑥 ∈ V → (♯‘𝑥) ∈ ℝ*) | |
22 | 21 | elv 3450 | . . . . 5 ⊢ (♯‘𝑥) ∈ ℝ* |
23 | 1lt2 12329 | . . . . . 6 ⊢ 1 < 2 | |
24 | 2re 12232 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
25 | 24 | rexri 11218 | . . . . . . 7 ⊢ 2 ∈ ℝ* |
26 | xrltletr 13082 | . . . . . . 7 ⊢ ((1 ∈ ℝ* ∧ 2 ∈ ℝ* ∧ (♯‘𝑥) ∈ ℝ*) → ((1 < 2 ∧ 2 ≤ (♯‘𝑥)) → 1 < (♯‘𝑥))) | |
27 | 20, 25, 22, 26 | mp3an 1462 | . . . . . 6 ⊢ ((1 < 2 ∧ 2 ≤ (♯‘𝑥)) → 1 < (♯‘𝑥)) |
28 | 23, 27 | mpan 689 | . . . . 5 ⊢ (2 ≤ (♯‘𝑥) → 1 < (♯‘𝑥)) |
29 | xrltne 13088 | . . . . 5 ⊢ ((1 ∈ ℝ* ∧ (♯‘𝑥) ∈ ℝ* ∧ 1 < (♯‘𝑥)) → (♯‘𝑥) ≠ 1) | |
30 | 20, 22, 28, 29 | mp3an12i 1466 | . . . 4 ⊢ (2 ≤ (♯‘𝑥) → (♯‘𝑥) ≠ 1) |
31 | 30 | ralimi 3083 | . . 3 ⊢ (∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥) → ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1) |
32 | 19, 31 | impbid1 224 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 {crab 3406 Vcvv 3444 ∖ cdif 3908 ∅c0 4283 𝒫 cpw 4561 {csn 4587 class class class wbr 5106 dom cdm 5634 ⟶wf 6493 ‘cfv 6497 0cc0 11056 1c1 11057 ℝ*cxr 11193 < clt 11194 ≤ cle 11195 2c2 12213 ℕ0*cxnn0 12490 ♯chash 14236 Vtxcvtx 27989 iEdgciedg 27990 Edgcedg 28040 UHGraphcuhgr 28049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-xnn0 12491 df-z 12505 df-uz 12769 df-fz 13431 df-hash 14237 df-edg 28041 df-uhgr 28051 |
This theorem is referenced by: lfuhgr3 33770 |
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