Theorem dfin5 3913

Description: Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)

Assertion

Ref	Expression
dfin5	⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfin5

Step	Hyp	Ref	Expression
1		df-in 3912	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
2		df-rab 3397	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
3	1, 2	eqtr4i 2755	1 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  {crab 3396   ∩ cin 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-rab 3397  df-in 3912

This theorem is referenced by:  incom  4162  ineq1  4166  nfinOLD  4178  rabbi2dva  4179  dfss7  4204  dfepfr  5607  epfrc  5608  pmtrmvd  19354  ablfaclem3  19987  mretopd  22996  ptclsg  23519  xkopt  23559  iscmet3  25210  xrlimcnp  26895  ppiub  27132  xppreima  32607  fpwrelmapffs  32696  orvcelval  34456  sstotbnd2  37773  glbconN  39375  glbconNOLD  39376  2polssN  39914  rfovcnvf1od  43997  fsovcnvlem  44006  ntrneifv3  44075  ntrneifv4  44078  clsneifv3  44103  clsneifv4  44104  neicvgfv  44114  inpw  48829