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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 9035 | . . . . 5 ⊢ (¬ 𝑉 ∈ Fin → ∃𝑛 𝑛 ∈ 𝑉) | |
2 | cusgrfi.v | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | eqeq1 2742 | . . . . . . . . . . . 12 ⊢ (𝑒 = 𝑝 → (𝑒 = {𝑣, 𝑛} ↔ 𝑝 = {𝑣, 𝑛})) | |
4 | 3 | anbi2d 629 | . . . . . . . . . . 11 ⊢ (𝑒 = 𝑝 → ((𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛}) ↔ (𝑣 ≠ 𝑛 ∧ 𝑝 = {𝑣, 𝑛}))) |
5 | 4 | rexbidv 3224 | . . . . . . . . . 10 ⊢ (𝑒 = 𝑝 → (∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛}) ↔ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑝 = {𝑣, 𝑛}))) |
6 | 5 | cbvrabv 3424 | . . . . . . . . 9 ⊢ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑝 = {𝑣, 𝑛})} |
7 | eqid 2738 | . . . . . . . . 9 ⊢ (𝑝 ∈ (𝑉 ∖ {𝑛}) ↦ {𝑝, 𝑛}) = (𝑝 ∈ (𝑉 ∖ {𝑛}) ↦ {𝑝, 𝑛}) | |
8 | 2, 6, 7 | cusgrfilem3 27812 | . . . . . . . 8 ⊢ (𝑛 ∈ 𝑉 → (𝑉 ∈ Fin ↔ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin)) |
9 | 8 | notbid 318 | . . . . . . 7 ⊢ (𝑛 ∈ 𝑉 → (¬ 𝑉 ∈ Fin ↔ ¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin)) |
10 | 9 | biimpac 479 | . . . . . 6 ⊢ ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → ¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin) |
11 | 2, 6 | cusgrfilem1 27810 | . . . . . . . . . 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 2829 | . . . . . . . . . . . 12 ⊢ (𝐸 ∈ Fin ↔ (Edg‘𝐺) ∈ Fin) |
14 | ssfi 8944 | . . . . . . . . . . . . 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 414 | . . . . . . . . . . . 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 241 | . . . . . . . . . . 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 152 | . . . . . . . . . 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 414 | . . . . . . . 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 482 | . . . . . 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 1938 | . . . 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 407 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝐸 ∈ Fin) → 𝑉 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 {crab 3068 ∖ cdif 3884 ⊆ wss 3887 𝒫 cpw 4534 {csn 4562 {cpr 4564 ↦ cmpt 5157 ‘cfv 6427 Fincfn 8721 Vtxcvtx 27354 Edgcedg 27405 ComplUSGraphccusgr 27765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-2o 8286 df-oadd 8289 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-dju 9647 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-2 12024 df-n0 12222 df-xnn0 12294 df-z 12308 df-uz 12571 df-fz 13228 df-hash 14033 df-edg 27406 df-upgr 27440 df-umgr 27441 df-usgr 27509 df-nbgr 27688 df-uvtx 27741 df-cplgr 27766 df-cusgr 27767 |
This theorem is referenced by: sizusglecusglem2 27817 |
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