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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 2821 | . . 3 ⊢ (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) | |
2 | simpllr 774 | . . 3 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑋) | |
3 | simplrl 775 | . . . 4 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑉) | |
4 | simpl 485 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → 𝑉 ∈ 𝑊) | |
5 | 4 | adantr 483 | . . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑉 ∈ 𝑊) |
6 | snex 5318 | . . . . . 6 ⊢ {〈𝐴, {𝐵, 𝐶}〉} ∈ V | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝐵 ≠ 𝐶 → {〈𝐴, {𝐵, 𝐶}〉} ∈ V) |
8 | opvtxfv 26775 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝐵, 𝐶}〉} ∈ V) → (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = 𝑉) | |
9 | 5, 7, 8 | syl2an 597 | . . . 4 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = 𝑉) |
10 | 3, 9 | eleqtrrd 2916 | . . 3 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉)) |
11 | simprr 771 | . . . . 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 597 | . . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = 𝑉) |
14 | 11, 13 | eleqtrrd 2916 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉)) |
15 | 14 | adantr 483 | . . 3 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (Vtx‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉)) |
16 | opiedgfv 26778 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝐵, 𝐶}〉} ∈ V) → (iEdg‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = {〈𝐴, {𝐵, 𝐶}〉}) | |
17 | 5, 7, 16 | syl2an 597 | . . 3 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → (iEdg‘〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉) = {〈𝐴, {𝐵, 𝐶}〉}) |
18 | simpr 487 | . . 3 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶) | |
19 | 1, 2, 10, 15, 17, 18 | usgr1e 27013 | . 2 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝐵 ≠ 𝐶) → 〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉 ∈ USGraph) |
20 | 19 | ex 415 | 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 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3486 {csn 4553 {cpr 4555 〈cop 4559 ‘cfv 6341 Vtxcvtx 26767 iEdgciedg 26768 USGraphcusgr 26920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-dju 9316 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-n0 11885 df-xnn0 11955 df-z 11969 df-uz 12231 df-fz 12883 df-hash 13681 df-vtx 26769 df-iedg 26770 df-edg 26819 df-uspgr 26921 df-usgr 26922 |
This theorem is referenced by: usgr2v1e2w 27020 |
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