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Mirrors > Home > MPE Home > Th. List > infi | Structured version Visualization version GIF version |
Description: The intersection of two sets is finite if one of them is. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
Ref | Expression |
---|---|
infi | ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐵) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4205 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | ssfi 8732 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐴 ∩ 𝐵) ∈ Fin) | |
3 | 1, 2 | mpan2 689 | 1 ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐵) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∩ cin 3935 ⊆ wss 3936 Fincfn 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-om 7575 df-er 8283 df-en 8504 df-fin 8507 |
This theorem is referenced by: rabfi 8737 resfnfinfin 8798 resfifsupp 8855 fin23lem22 9743 pmatcoe1fsupp 21303 trlsegvdeglem6 27998 gsummptres 30685 indsumin 31276 eulerpartlemt 31624 ballotlemgun 31777 hgt750lemd 31914 fourierdlem50 42434 fourierdlem71 42455 fourierdlem76 42460 fourierdlem80 42464 fourierdlem103 42487 fourierdlem104 42488 sge0split 42684 |
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