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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 4053 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | ssfi 8470 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐴 ∩ 𝐵) ∈ Fin) | |
3 | 1, 2 | mpan2 681 | 1 ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐵) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∩ cin 3791 ⊆ wss 3792 Fincfn 8243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-om 7346 df-er 8028 df-en 8244 df-fin 8247 |
This theorem is referenced by: rabfi 8475 resfnfinfin 8536 resfifsupp 8593 fin23lem22 9486 pmatcoe1fsupp 20917 trlsegvdeglem6 27633 gsummptres 30350 indsumin 30686 eulerpartlemt 31035 ballotlemgun 31189 hgt750lemd 31332 fourierdlem50 41310 fourierdlem71 41331 fourierdlem76 41336 fourierdlem80 41340 fourierdlem103 41363 fourierdlem104 41364 sge0split 41560 |
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