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Mirrors > Home > MPE Home > Th. List > umgredg | Structured version Visualization version GIF version |
Description: For each edge in a multigraph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
upgredg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upgredg.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
umgredg | ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgredg.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
2 | 1 | eleq2i 2722 | . . . 4 ⊢ (𝐶 ∈ 𝐸 ↔ 𝐶 ∈ (Edg‘𝐺)) |
3 | edgumgr 26075 | . . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ (Edg‘𝐺)) → (𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐶) = 2)) | |
4 | 2, 3 | sylan2b 491 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ 𝐸) → (𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐶) = 2)) |
5 | hash2prde 13290 | . . . 4 ⊢ ((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐶) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏})) | |
6 | eleq1 2718 | . . . . . . . . . 10 ⊢ (𝐶 = {𝑎, 𝑏} → (𝐶 ∈ 𝒫 (Vtx‘𝐺) ↔ {𝑎, 𝑏} ∈ 𝒫 (Vtx‘𝐺))) | |
7 | prex 4939 | . . . . . . . . . . . 12 ⊢ {𝑎, 𝑏} ∈ V | |
8 | 7 | elpw 4197 | . . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝒫 (Vtx‘𝐺) ↔ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) |
9 | vex 3234 | . . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V | |
10 | vex 3234 | . . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V | |
11 | 9, 10 | prss 4383 | . . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ {𝑎, 𝑏} ⊆ 𝑉) |
12 | upgredg.v | . . . . . . . . . . . . 13 ⊢ 𝑉 = (Vtx‘𝐺) | |
13 | 12 | sseq2i 3663 | . . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ⊆ 𝑉 ↔ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) |
14 | 11, 13 | sylbbr 226 | . . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ⊆ (Vtx‘𝐺) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) |
15 | 8, 14 | sylbi 207 | . . . . . . . . . 10 ⊢ ({𝑎, 𝑏} ∈ 𝒫 (Vtx‘𝐺) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) |
16 | 6, 15 | syl6bi 243 | . . . . . . . . 9 ⊢ (𝐶 = {𝑎, 𝑏} → (𝐶 ∈ 𝒫 (Vtx‘𝐺) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) |
17 | 16 | adantrd 483 | . . . . . . . 8 ⊢ (𝐶 = {𝑎, 𝑏} → ((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐶) = 2) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) |
18 | 17 | adantl 481 | . . . . . . 7 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}) → ((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐶) = 2) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) |
19 | 18 | imdistanri 727 | . . . . . 6 ⊢ (((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐶) = 2) ∧ (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏})) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}))) |
20 | 19 | ex 449 | . . . . 5 ⊢ ((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐶) = 2) → ((𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏})))) |
21 | 20 | 2eximdv 1888 | . . . 4 ⊢ ((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐶) = 2) → (∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}) → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏})))) |
22 | 5, 21 | mpd 15 | . . 3 ⊢ ((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐶) = 2) → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}))) |
23 | 4, 22 | syl 17 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}))) |
24 | r2ex 3090 | . 2 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}) ↔ ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}))) | |
25 | 23, 24 | sylibr 224 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∃wex 1744 ∈ wcel 2030 ≠ wne 2823 ∃wrex 2942 ⊆ wss 3607 𝒫 cpw 4191 {cpr 4212 ‘cfv 5926 2c2 11108 #chash 13157 Vtxcvtx 25919 Edgcedg 25984 UMGraphcumgr 26021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-hash 13158 df-edg 25985 df-umgr 26023 |
This theorem is referenced by: usgredg 26136 |
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