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Mirrors > Home > MPE Home > Th. List > umgrnloopv | Structured version Visualization version GIF version |
Description: In a multigraph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 11-Dec-2020.) |
Ref | Expression |
---|---|
umgrnloopv.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
umgrnloopv | ⊢ ((𝐺 ∈ UMGraph ∧ 𝑀 ∈ 𝑊) → ((𝐸‘𝑋) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prnzg 4674 | . . . . . . . 8 ⊢ (𝑀 ∈ 𝑊 → {𝑀, 𝑁} ≠ ∅) | |
2 | 1 | adantl 485 | . . . . . . 7 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → {𝑀, 𝑁} ≠ ∅) |
3 | neeq1 3049 | . . . . . . . 8 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → ((𝐸‘𝑋) ≠ ∅ ↔ {𝑀, 𝑁} ≠ ∅)) | |
4 | 3 | adantr 484 | . . . . . . 7 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → ((𝐸‘𝑋) ≠ ∅ ↔ {𝑀, 𝑁} ≠ ∅)) |
5 | 2, 4 | mpbird 260 | . . . . . 6 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → (𝐸‘𝑋) ≠ ∅) |
6 | fvfundmfvn0 6683 | . . . . . 6 ⊢ ((𝐸‘𝑋) ≠ ∅ → (𝑋 ∈ dom 𝐸 ∧ Fun (𝐸 ↾ {𝑋}))) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → (𝑋 ∈ dom 𝐸 ∧ Fun (𝐸 ↾ {𝑋}))) |
8 | eqid 2798 | . . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
9 | umgrnloopv.e | . . . . . . . . . 10 ⊢ 𝐸 = (iEdg‘𝐺) | |
10 | 8, 9 | umgredg2 26893 | . . . . . . . . 9 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸‘𝑋)) = 2) |
11 | fveqeq2 6654 | . . . . . . . . . . 11 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → ((♯‘(𝐸‘𝑋)) = 2 ↔ (♯‘{𝑀, 𝑁}) = 2)) | |
12 | eqid 2798 | . . . . . . . . . . . . 13 ⊢ {𝑀, 𝑁} = {𝑀, 𝑁} | |
13 | 12 | hashprdifel 13755 | . . . . . . . . . . . 12 ⊢ ((♯‘{𝑀, 𝑁}) = 2 → (𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁} ∧ 𝑀 ≠ 𝑁)) |
14 | 13 | simp3d 1141 | . . . . . . . . . . 11 ⊢ ((♯‘{𝑀, 𝑁}) = 2 → 𝑀 ≠ 𝑁) |
15 | 11, 14 | syl6bi 256 | . . . . . . . . . 10 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → ((♯‘(𝐸‘𝑋)) = 2 → 𝑀 ≠ 𝑁)) |
16 | 15 | adantr 484 | . . . . . . . . 9 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → ((♯‘(𝐸‘𝑋)) = 2 → 𝑀 ≠ 𝑁)) |
17 | 10, 16 | syl5com 31 | . . . . . . . 8 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸) → (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → 𝑀 ≠ 𝑁)) |
18 | 17 | expcom 417 | . . . . . . 7 ⊢ (𝑋 ∈ dom 𝐸 → (𝐺 ∈ UMGraph → (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → 𝑀 ≠ 𝑁))) |
19 | 18 | com23 86 | . . . . . 6 ⊢ (𝑋 ∈ dom 𝐸 → (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → (𝐺 ∈ UMGraph → 𝑀 ≠ 𝑁))) |
20 | 19 | adantr 484 | . . . . 5 ⊢ ((𝑋 ∈ dom 𝐸 ∧ Fun (𝐸 ↾ {𝑋})) → (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → (𝐺 ∈ UMGraph → 𝑀 ≠ 𝑁))) |
21 | 7, 20 | mpcom 38 | . . . 4 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → (𝐺 ∈ UMGraph → 𝑀 ≠ 𝑁)) |
22 | 21 | ex 416 | . . 3 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → (𝑀 ∈ 𝑊 → (𝐺 ∈ UMGraph → 𝑀 ≠ 𝑁))) |
23 | 22 | com13 88 | . 2 ⊢ (𝐺 ∈ UMGraph → (𝑀 ∈ 𝑊 → ((𝐸‘𝑋) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁))) |
24 | 23 | imp 410 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑀 ∈ 𝑊) → ((𝐸‘𝑋) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 {csn 4525 {cpr 4527 dom cdm 5519 ↾ cres 5521 Fun wfun 6318 ‘cfv 6324 2c2 11680 ♯chash 13686 Vtxcvtx 26789 iEdgciedg 26790 UMGraphcumgr 26874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-umgr 26876 |
This theorem is referenced by: umgrnloop 26901 usgrnloopv 26990 |
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