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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 27297 | . . . . 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 1102 | . . . . . . . 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 1161 | . . . . . . . 8 ⊢ ((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦) ∧ 𝑥 ≠ 𝑦) ↔ (𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦) ∧ ¬ 𝑥 = 𝑦)) |
7 | 4, 6 | bitri 278 | . . . . . . 7 ⊢ ((𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ↔ (𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦) ∧ ¬ 𝑥 = 𝑦)) |
8 | df-3an 1091 | . . . . . . 7 ⊢ ((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ∧ ¬ 𝑥 = 𝑦)) | |
9 | 7, 8 | bitri 278 | . . . . . 6 ⊢ ((𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ↔ ((𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)) ∧ ¬ 𝑥 = 𝑦)) |
10 | 9 | 2rexbii 3171 | . . . . 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 221 | . . . 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 3184 | . . . . . 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 3170 | . . . . 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 3160 | . . . . 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 278 | . . . 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 221 | . . 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 492 | . 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 6717 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐼‘𝑥) = (𝐼‘𝑦)) | |
19 | 18 | eleq2d 2823 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑁 ∈ (𝐼‘𝑥) ↔ 𝑁 ∈ (𝐼‘𝑦))) |
20 | 19 | reu4 3644 | . 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 332 | 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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 ∃wrex 3062 ∃!wreu 3063 {cpr 4543 dom cdm 5551 ‘cfv 6380 iEdgciedg 27088 Edgcedg 27138 UMGraphcumgr 27172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-hash 13897 df-edg 27139 df-uhgr 27149 df-upgr 27173 df-umgr 27174 |
This theorem is referenced by: usgr2edg1 27300 |
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