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Mirrors > Home > MPE Home > Th. List > usgr1eop | Structured version Visualization version GIF version |
Description: A simple graph with (at least) two different vertices and one edge. If the two vertices were not different, the edge would be a loop. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.) |
Ref | Expression |
---|---|
usgr1eop | ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ≠ 𝐶 → 〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉 ∈ USGraph)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) | |
2 | simpllr 776 | . . 3 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑋) | |
3 | simplrl 777 | . . . 4 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑉) | |
4 | simpl 486 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → 𝑉 ∈ 𝑊) | |
5 | 4 | adantr 484 | . . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑉 ∈ 𝑊) |
6 | snex 5324 | . . . . . 6 ⊢ {〈𝐴, {𝐵, 𝐶}〉} ∈ V | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝐵 ≠ 𝐶 → {〈𝐴, {𝐵, 𝐶}〉} ∈ V) |
8 | opvtxfv 27095 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝐵, 𝐶}〉} ∈ V) → (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = 𝑉) | |
9 | 5, 7, 8 | syl2an 599 | . . . 4 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = 𝑉) |
10 | 3, 9 | eleqtrrd 2841 | . . 3 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉)) |
11 | simprr 773 | . . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
12 | 6 | a1i 11 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {〈𝐴, {𝐵, 𝐶}〉} ∈ V) |
13 | 4, 12, 8 | syl2an 599 | . . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = 𝑉) |
14 | 11, 13 | eleqtrrd 2841 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉)) |
15 | 14 | adantr 484 | . . 3 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉)) |
16 | opiedgfv 27098 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝐵, 𝐶}〉} ∈ V) → (iEdg‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = {〈𝐴, {𝐵, 𝐶}〉}) | |
17 | 5, 7, 16 | syl2an 599 | . . 3 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → (iEdg‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = {〈𝐴, {𝐵, 𝐶}〉}) |
18 | simpr 488 | . . 3 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶) | |
19 | 1, 2, 10, 15, 17, 18 | usgr1e 27333 | . 2 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉 ∈ USGraph) |
20 | 19 | ex 416 | 1 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ≠ 𝐶 → 〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉 ∈ USGraph)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 Vcvv 3408 {csn 4541 {cpr 4543 〈cop 4547 ‘cfv 6380 Vtxcvtx 27087 iEdgciedg 27088 USGraphcusgr 27240 |
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-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-xnn0 12163 df-z 12177 df-uz 12439 df-fz 13096 df-hash 13897 df-vtx 27089 df-iedg 27090 df-edg 27139 df-uspgr 27241 df-usgr 27242 |
This theorem is referenced by: usgr2v1e2w 27340 |
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