Theorem elfpw 9264

Description: Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
elfpw	⊢ (𝐴 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ Fin))

Proof of Theorem elfpw

Step	Hyp	Ref	Expression
1		elin 3905	. 2 ⊢ (𝐴 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝐴 ∈ 𝒫 𝐵 ∧ 𝐴 ∈ Fin))
2		elpwg 4544	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
3	2	pm5.32ri 575	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ 𝐴 ∈ Fin) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ Fin))
4	1, 3	bitri 275	1 ⊢ (𝐴 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ Fin))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541  Fincfn 8893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-in 3896  df-ss 3906  df-pw 4543

This theorem is referenced by:  bitsinv2  16412  bitsf1ocnv  16413  2ebits  16416  bitsinvp1  16418  sadcaddlem  16426  sadadd2lem  16428  sadadd3  16430  sadaddlem  16435  sadasslem  16439  sadeq  16441  firest  17395  acsfiindd  18519  restfpw  23144  cmpcov2  23355  cmpcovf  23356  cncmp  23357  tgcmp  23366  cmpcld  23367  cmpfi  23373  locfincmp  23491  comppfsc  23497  alexsublem  24009  alexsubALTlem2  24013  alexsubALTlem4  24015  alexsubALT  24016  ptcmplem2  24018  ptcmplem3  24019  ptcmplem5  24021  tsmsfbas  24093  tsmslem1  24094  tsmsgsum  24104  tsmssubm  24108  tsmsres  24109  tsmsf1o  24110  tsmsmhm  24111  tsmsadd  24112  tsmsxplem1  24118  tsmsxplem2  24119  tsmsxp  24120  xrge0gsumle  24799  xrge0tsms  24800  indf1ofs  32926  xrge0tsmsd  33134  mvrsfpw  35688  elmpst  35718  istotbnd3  38092  sstotbnd2  38095  sstotbnd  38096  sstotbnd3  38097  equivtotbnd  38099  totbndbnd  38110  prdstotbnd  38115  isnacs3  43142  pwfi2f1o  43524  hbtlem6  43557