Theorem dfin5 3922

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 3921	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
2		df-rab 3406	. 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 3405   ∩ cin 3913

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 3406  df-in 3921

This theorem is referenced by:  incom  4172  ineq1  4176  nfinOLD  4188  rabbi2dva  4189  dfss7  4214  dfepfr  5622  epfrc  5623  pmtrmvd  19386  ablfaclem3  20019  mretopd  22979  ptclsg  23502  xkopt  23542  iscmet3  25193  xrlimcnp  26878  ppiub  27115  xppreima  32569  fpwrelmapffs  32657  orvcelval  34460  sstotbnd2  37768  glbconN  39370  glbconNOLD  39371  2polssN  39909  rfovcnvf1od  43993  fsovcnvlem  44002  ntrneifv3  44071  ntrneifv4  44074  clsneifv3  44099  clsneifv4  44100  neicvgfv  44110  inpw  48813