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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 3795 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | ssfi 8065 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐴 ∩ 𝐵) ∈ Fin) | |
3 | 1, 2 | mpan2 703 | 1 ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐵) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-om 6958 df-er 7629 df-en 7842 df-fin 7845 |
This theorem is referenced by: rabfi 8070 resfifsupp 8186 fin23lem22 9032 pmatcoe1fsupp 20325 gsummptres 29115 eulerpartlemt 29760 ballotlemgun 29913 fourierdlem50 39049 fourierdlem71 39070 fourierdlem76 39075 fourierdlem80 39079 fourierdlem103 39102 fourierdlem104 39103 sge0split 39302 resfnfinfin 40339 trlsegvdeglem6 41393 |
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