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Mirrors > Home > MPE Home > Th. List > 1hegrvtxdg1r | Structured version Visualization version GIF version |
Description: The vertex degree of a graph with one hyperedge, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 23-Feb-2021.) |
Ref | Expression |
---|---|
1hegrvtxdg1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
1hegrvtxdg1.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
1hegrvtxdg1.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
1hegrvtxdg1.n | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
1hegrvtxdg1.x | ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) |
1hegrvtxdg1.i | ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) |
1hegrvtxdg1.e | ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) |
1hegrvtxdg1.v | ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) |
Ref | Expression |
---|---|
1hegrvtxdg1r | ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1hegrvtxdg1.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
2 | 1hegrvtxdg1.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
3 | 1hegrvtxdg1.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | 1hegrvtxdg1.n | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
5 | 4 | necomd 2998 | . 2 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
6 | 1hegrvtxdg1.x | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) | |
7 | 1hegrvtxdg1.i | . 2 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) | |
8 | prcom 4404 | . . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶} | |
9 | 1hegrvtxdg1.e | . . 3 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) | |
10 | 8, 9 | syl5eqss 3799 | . 2 ⊢ (𝜑 → {𝐶, 𝐵} ⊆ 𝐸) |
11 | 1hegrvtxdg1.v | . 2 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | |
12 | 1, 2, 3, 5, 6, 7, 10, 11 | 1hegrvtxdg1 26639 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ⊆ wss 3724 𝒫 cpw 4298 {csn 4317 {cpr 4319 〈cop 4323 ‘cfv 6032 1c1 10140 Vtxcvtx 26096 iEdgciedg 26097 VtxDegcvtxdg 26597 |
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 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 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 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-1st 7316 df-2nd 7317 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-oadd 7718 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-fin 8114 df-card 8966 df-cda 9193 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-nn 11224 df-2 11282 df-n0 11496 df-xnn0 11567 df-z 11581 df-uz 11890 df-xadd 12153 df-fz 12535 df-hash 13323 df-vtxdg 26598 |
This theorem is referenced by: eupth2lem3lem4 27412 |
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