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Mirrors > Home > MPE Home > Th. List > uspgr1v1eop | Structured version Visualization version GIF version |
Description: A simple pseudograph with (at least) one vertex and one edge (a loop). (Contributed by AV, 5-Dec-2020.) |
Ref | Expression |
---|---|
uspgr1v1eop | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑉) → 〈𝑉, {〈𝐴, {𝐵}〉}〉 ∈ USPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4329 | . . . . 5 ⊢ {𝐵} = {𝐵, 𝐵} | |
2 | 1 | opeq2i 4543 | . . . 4 ⊢ 〈𝐴, {𝐵}〉 = 〈𝐴, {𝐵, 𝐵}〉 |
3 | 2 | sneqi 4327 | . . 3 ⊢ {〈𝐴, {𝐵}〉} = {〈𝐴, {𝐵, 𝐵}〉} |
4 | 3 | opeq2i 4543 | . 2 ⊢ 〈𝑉, {〈𝐴, {𝐵}〉}〉 = 〈𝑉, {〈𝐴, {𝐵, 𝐵}〉}〉 |
5 | 3simpa 1142 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑉) → (𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋)) | |
6 | id 22 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ 𝑉) | |
7 | 6 | ancri 539 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) |
8 | 7 | 3ad2ant3 1129 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) |
9 | uspgr1eop 26355 | . . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 〈𝑉, {〈𝐴, {𝐵, 𝐵}〉}〉 ∈ USPGraph) | |
10 | 5, 8, 9 | syl2anc 573 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑉) → 〈𝑉, {〈𝐴, {𝐵, 𝐵}〉}〉 ∈ USPGraph) |
11 | 4, 10 | syl5eqel 2854 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑉) → 〈𝑉, {〈𝐴, {𝐵}〉}〉 ∈ USPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 ∈ wcel 2145 {csn 4316 {cpr 4318 〈cop 4322 USPGraphcuspgr 26258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-card 8963 df-cda 9190 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-nn 11221 df-2 11279 df-n0 11493 df-xnn0 11564 df-z 11578 df-uz 11887 df-fz 12527 df-hash 13315 df-vtx 26090 df-iedg 26091 df-uspgr 26260 |
This theorem is referenced by: (None) |
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