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Mirrors > Home > MPE Home > Th. List > cusgrfi | Structured version Visualization version GIF version |
Description: If the size of a complete simple graph is finite, then its order is also finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
Ref | Expression |
---|---|
cusgrfi.v | ⊢ 𝑉 = (Vtx‘𝐺) |
cusgrfi.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
cusgrfi | ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝐸 ∈ Fin) → 𝑉 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfielex 8541 | . . . . 5 ⊢ (¬ 𝑉 ∈ Fin → ∃𝑛 𝑛 ∈ 𝑉) | |
2 | cusgrfi.v | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | eqeq1 2777 | . . . . . . . . . . . 12 ⊢ (𝑒 = 𝑝 → (𝑒 = {𝑣, 𝑛} ↔ 𝑝 = {𝑣, 𝑛})) | |
4 | 3 | anbi2d 620 | . . . . . . . . . . 11 ⊢ (𝑒 = 𝑝 → ((𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛}) ↔ (𝑣 ≠ 𝑛 ∧ 𝑝 = {𝑣, 𝑛}))) |
5 | 4 | rexbidv 3237 | . . . . . . . . . 10 ⊢ (𝑒 = 𝑝 → (∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛}) ↔ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑝 = {𝑣, 𝑛}))) |
6 | 5 | cbvrabv 3407 | . . . . . . . . 9 ⊢ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑝 = {𝑣, 𝑛})} |
7 | eqid 2773 | . . . . . . . . 9 ⊢ (𝑝 ∈ (𝑉 ∖ {𝑛}) ↦ {𝑝, 𝑛}) = (𝑝 ∈ (𝑉 ∖ {𝑛}) ↦ {𝑝, 𝑛}) | |
8 | 2, 6, 7 | cusgrfilem3 26958 | . . . . . . . 8 ⊢ (𝑛 ∈ 𝑉 → (𝑉 ∈ Fin ↔ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin)) |
9 | 8 | notbid 310 | . . . . . . 7 ⊢ (𝑛 ∈ 𝑉 → (¬ 𝑉 ∈ Fin ↔ ¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin)) |
10 | 9 | biimpac 471 | . . . . . 6 ⊢ ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → ¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin) |
11 | 2, 6 | cusgrfilem1 26956 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑛 ∈ 𝑉) → {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ⊆ (Edg‘𝐺)) |
12 | cusgrfi.e | . . . . . . . . . . . . 13 ⊢ 𝐸 = (Edg‘𝐺) | |
13 | 12 | eleq1i 2851 | . . . . . . . . . . . 12 ⊢ (𝐸 ∈ Fin ↔ (Edg‘𝐺) ∈ Fin) |
14 | ssfi 8532 | . . . . . . . . . . . . 13 ⊢ (((Edg‘𝐺) ∈ Fin ∧ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ⊆ (Edg‘𝐺)) → {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin) | |
15 | 14 | expcom 406 | . . . . . . . . . . . 12 ⊢ ({𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ⊆ (Edg‘𝐺) → ((Edg‘𝐺) ∈ Fin → {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin)) |
16 | 13, 15 | syl5bi 234 | . . . . . . . . . . 11 ⊢ ({𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ⊆ (Edg‘𝐺) → (𝐸 ∈ Fin → {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin)) |
17 | 16 | con3d 150 | . . . . . . . . . 10 ⊢ ({𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ⊆ (Edg‘𝐺) → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → ¬ 𝐸 ∈ Fin)) |
18 | 11, 17 | syl 17 | . . . . . . . . 9 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑛 ∈ 𝑉) → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → ¬ 𝐸 ∈ Fin)) |
19 | 18 | expcom 406 | . . . . . . . 8 ⊢ (𝑛 ∈ 𝑉 → (𝐺 ∈ ComplUSGraph → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → ¬ 𝐸 ∈ Fin))) |
20 | 19 | com23 86 | . . . . . . 7 ⊢ (𝑛 ∈ 𝑉 → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → (𝐺 ∈ ComplUSGraph → ¬ 𝐸 ∈ Fin))) |
21 | 20 | adantl 474 | . . . . . 6 ⊢ ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → (𝐺 ∈ ComplUSGraph → ¬ 𝐸 ∈ Fin))) |
22 | 10, 21 | mpd 15 | . . . . 5 ⊢ ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → (𝐺 ∈ ComplUSGraph → ¬ 𝐸 ∈ Fin)) |
23 | 1, 22 | exlimddv 1895 | . . . 4 ⊢ (¬ 𝑉 ∈ Fin → (𝐺 ∈ ComplUSGraph → ¬ 𝐸 ∈ Fin)) |
24 | 23 | com12 32 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (¬ 𝑉 ∈ Fin → ¬ 𝐸 ∈ Fin)) |
25 | 24 | con4d 115 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → (𝐸 ∈ Fin → 𝑉 ∈ Fin)) |
26 | 25 | imp 398 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝐸 ∈ Fin) → 𝑉 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 ∃wrex 3084 {crab 3087 ∖ cdif 3821 ⊆ wss 3824 𝒫 cpw 4417 {csn 4436 {cpr 4438 ↦ cmpt 5005 ‘cfv 6186 Fincfn 8305 Vtxcvtx 26500 Edgcedg 26551 ComplUSGraphccusgr 26911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-dju 9123 df-card 9161 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-n0 11707 df-xnn0 11779 df-z 11793 df-uz 12058 df-fz 12708 df-hash 13505 df-edg 26552 df-upgr 26586 df-umgr 26587 df-usgr 26655 df-nbgr 26834 df-uvtx 26887 df-cplgr 26912 df-cusgr 26913 |
This theorem is referenced by: sizusglecusglem2 26963 |
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