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Theorem 0pwfi 42168
Description: The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
0pwfi ∅ ∈ (𝒫 𝐴 ∩ Fin)

Proof of Theorem 0pwfi
StepHypRef Expression
1 0elpw 5223 . 2 ∅ ∈ 𝒫 𝐴
2 0fin 8773 . 2 ∅ ∈ Fin
31, 2elini 4084 1 ∅ ∈ (𝒫 𝐴 ∩ Fin)
Colors of variables: wff setvar class
Syntax hints:  wcel 2114  cin 3843  c0 4212  𝒫 cpw 4489  Fincfn 8558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-om 7603  df-en 8559  df-fin 8562
This theorem is referenced by:  pwfin0  42171
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