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Mirrors > Home > MPE Home > Th. List > umgr2edg1 | Structured version Visualization version GIF version |
Description: If a vertex is adjacent to two different vertices in a multigraph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 8-Jun-2021.) |
Ref | Expression |
---|---|
usgrf1oedg.i | ⊢ 𝐼 = (iEdg‘𝐺) |
usgrf1oedg.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
umgr2edg1 | ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrf1oedg.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
2 | usgrf1oedg.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | umgr2edg 28463 | . . . . 5 ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦))) |
4 | 3anrot 1100 | . . . . . . . 8 ⊢ ((𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ↔ (𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦) ∧ 𝑥 ≠ 𝑦)) | |
5 | df-ne 2941 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | |
6 | 5 | 3anbi3i 1159 | . . . . . . . 8 ⊢ ((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦) ∧ 𝑥 ≠ 𝑦) ↔ (𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦) ∧ ¬ 𝑥 = 𝑦)) |
7 | 4, 6 | bitri 274 | . . . . . . 7 ⊢ ((𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ↔ (𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦) ∧ ¬ 𝑥 = 𝑦)) |
8 | df-3an 1089 | . . . . . . 7 ⊢ ((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ∧ ¬ 𝑥 = 𝑦)) | |
9 | 7, 8 | bitri 274 | . . . . . 6 ⊢ ((𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ↔ ((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ∧ ¬ 𝑥 = 𝑦)) |
10 | 9 | 2rexbii 3129 | . . . . 5 ⊢ (∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ↔ ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ∧ ¬ 𝑥 = 𝑦)) |
11 | 3, 10 | sylib 217 | . . . 4 ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ∧ ¬ 𝑥 = 𝑦)) |
12 | rexanali 3102 | . . . . . 6 ⊢ (∃𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ ∀𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) → 𝑥 = 𝑦)) | |
13 | 12 | rexbii 3094 | . . . . 5 ⊢ (∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑥 ∈ dom 𝐼 ¬ ∀𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) → 𝑥 = 𝑦)) |
14 | rexnal 3100 | . . . . 5 ⊢ (∃𝑥 ∈ dom 𝐼 ¬ ∀𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥 ∈ dom 𝐼∀𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) → 𝑥 = 𝑦)) | |
15 | 13, 14 | bitri 274 | . . . 4 ⊢ (∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ ∀𝑥 ∈ dom 𝐼∀𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) → 𝑥 = 𝑦)) |
16 | 11, 15 | sylib 217 | . . 3 ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∀𝑥 ∈ dom 𝐼∀𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) → 𝑥 = 𝑦)) |
17 | 16 | intnand 489 | . 2 ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ (∃𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼‘𝑥) ∧ ∀𝑥 ∈ dom 𝐼∀𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) → 𝑥 = 𝑦))) |
18 | fveq2 6891 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐼‘𝑥) = (𝐼‘𝑦)) | |
19 | 18 | eleq2d 2819 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑁 ∈ (𝐼‘𝑥) ↔ 𝑁 ∈ (𝐼‘𝑦))) |
20 | 19 | reu4 3727 | . 2 ⊢ (∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼‘𝑥) ↔ (∃𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼‘𝑥) ∧ ∀𝑥 ∈ dom 𝐼∀𝑦 ∈ dom 𝐼((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) → 𝑥 = 𝑦))) |
21 | 17, 20 | sylnibr 328 | 1 ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∃!wreu 3374 {cpr 4630 dom cdm 5676 ‘cfv 6543 iEdgciedg 28254 Edgcedg 28304 UMGraphcumgr 28338 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-hash 14290 df-edg 28305 df-uhgr 28315 df-upgr 28339 df-umgr 28340 |
This theorem is referenced by: usgr2edg1 28466 |
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