Theorem dfin5 3971

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 3970	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
2		df-rab 3434	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
3	1, 2	eqtr4i 2766	1 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  {crab 3433   ∩ cin 3962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-rab 3434  df-in 3970

This theorem is referenced by:  incom  4217  ineq1  4221  nfinOLD  4233  rabbi2dva  4234  dfss7  4257  dfepfr  5673  epfrc  5674  pmtrmvd  19489  ablfaclem3  20122  mretopd  23116  ptclsg  23639  xkopt  23679  iscmet3  25341  xrlimcnp  27026  ppiub  27263  xppreima  32662  fpwrelmapffs  32752  orvcelval  34450  sstotbnd2  37761  glbconN  39359  glbconNOLD  39360  2polssN  39898  rfovcnvf1od  43994  fsovcnvlem  44003  ntrneifv3  44072  ntrneifv4  44075  clsneifv3  44100  clsneifv4  44101  neicvgfv  44111  inpw  48667