Theorem elfpw 9238

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 3913	. 2 ⊢ (𝐴 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝐴 ∈ 𝒫 𝐵 ∧ 𝐴 ∈ Fin))
2		elpwg 4550	. . 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 2111   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547  Fincfn 8869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3904  df-ss 3914  df-pw 4549

This theorem is referenced by:  bitsinv2  16354  bitsf1ocnv  16355  2ebits  16358  bitsinvp1  16360  sadcaddlem  16368  sadadd2lem  16370  sadadd3  16372  sadaddlem  16377  sadasslem  16381  sadeq  16383  firest  17336  acsfiindd  18459  restfpw  23094  cmpcov2  23305  cmpcovf  23306  cncmp  23307  tgcmp  23316  cmpcld  23317  cmpfi  23323  locfincmp  23441  comppfsc  23447  alexsublem  23959  alexsubALTlem2  23963  alexsubALTlem4  23965  alexsubALT  23966  ptcmplem2  23968  ptcmplem3  23969  ptcmplem5  23971  tsmsfbas  24043  tsmslem1  24044  tsmsgsum  24054  tsmssubm  24058  tsmsres  24059  tsmsf1o  24060  tsmsmhm  24061  tsmsadd  24062  tsmsxplem1  24068  tsmsxplem2  24069  tsmsxp  24070  xrge0gsumle  24749  xrge0tsms  24750  indf1ofs  32847  xrge0tsmsd  33042  mvrsfpw  35550  elmpst  35580  istotbnd3  37810  sstotbnd2  37813  sstotbnd  37814  sstotbnd3  37815  equivtotbnd  37817  totbndbnd  37828  prdstotbnd  37833  isnacs3  42802  pwfi2f1o  43188  hbtlem6  43221