Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4202 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | ssfi 8726 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐴 ∩ 𝐵) ∈ Fin) | |
3 | 1, 2 | mpan2 687 | 1 ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐵) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 Fincfn 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-om 7570 df-er 8278 df-en 8498 df-fin 8501 |
This theorem is referenced by: rabfi 8731 resfnfinfin 8792 resfifsupp 8849 fin23lem22 9737 pmatcoe1fsupp 21237 trlsegvdeglem6 27931 gsummptres 30617 indsumin 31180 eulerpartlemt 31528 ballotlemgun 31681 hgt750lemd 31818 fourierdlem50 42318 fourierdlem71 42339 fourierdlem76 42344 fourierdlem80 42348 fourierdlem103 42371 fourierdlem104 42372 sge0split 42568 |
Copyright terms: Public domain | W3C validator |