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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 2829 | . . . 4 ⊢ (𝐶 ∈ 𝐸 ↔ 𝐶 ∈ (Edg‘𝐺)) |
3 | edgumgr 28033 | . . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ (Edg‘𝐺)) → (𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐶) = 2)) | |
4 | 2, 3 | sylan2b 594 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ 𝐸) → (𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐶) = 2)) |
5 | hash2prde 14368 | . . . 4 ⊢ ((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐶) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏})) | |
6 | eleq1 2825 | . . . . . . . . . 10 ⊢ (𝐶 = {𝑎, 𝑏} → (𝐶 ∈ 𝒫 (Vtx‘𝐺) ↔ {𝑎, 𝑏} ∈ 𝒫 (Vtx‘𝐺))) | |
7 | prex 5389 | . . . . . . . . . . . 12 ⊢ {𝑎, 𝑏} ∈ V | |
8 | 7 | elpw 4564 | . . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝒫 (Vtx‘𝐺) ↔ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) |
9 | vex 3449 | . . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V | |
10 | vex 3449 | . . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V | |
11 | 9, 10 | prss 4780 | . . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ {𝑎, 𝑏} ⊆ 𝑉) |
12 | upgredg.v | . . . . . . . . . . . . 13 ⊢ 𝑉 = (Vtx‘𝐺) | |
13 | 12 | sseq2i 3973 | . . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ⊆ 𝑉 ↔ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) |
14 | 11, 13 | sylbbr 235 | . . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ⊆ (Vtx‘𝐺) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) |
15 | 8, 14 | sylbi 216 | . . . . . . . . . 10 ⊢ ({𝑎, 𝑏} ∈ 𝒫 (Vtx‘𝐺) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) |
16 | 6, 15 | syl6bi 252 | . . . . . . . . 9 ⊢ (𝐶 = {𝑎, 𝑏} → (𝐶 ∈ 𝒫 (Vtx‘𝐺) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) |
17 | 16 | adantrd 492 | . . . . . . . 8 ⊢ (𝐶 = {𝑎, 𝑏} → ((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐶) = 2) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) |
18 | 17 | adantl 482 | . . . . . . 7 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}) → ((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐶) = 2) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) |
19 | 18 | imdistanri 570 | . . . . . 6 ⊢ (((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐶) = 2) ∧ (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏})) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}))) |
20 | 19 | ex 413 | . . . . 5 ⊢ ((𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐶) = 2) → ((𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏})))) |
21 | 20 | 2eximdv 1922 | . . . 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 3192 | . 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 233 | 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 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 ⊆ wss 3910 𝒫 cpw 4560 {cpr 4588 ‘cfv 6496 2c2 12207 ♯chash 14229 Vtxcvtx 27894 Edgcedg 27945 UMGraphcumgr 27979 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-dju 9836 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-n0 12413 df-z 12499 df-uz 12763 df-fz 13424 df-hash 14230 df-edg 27946 df-umgr 27981 |
This theorem is referenced by: usgredg 28094 umgr2cycllem 33674 |
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