Theorem dfin5 3136

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 3135	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
2		df-rab 2464	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
3	1, 2	eqtr4i 2201	1 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}

Syntax hints:   ∧ wa 104   = wceq 1353   ∈ wcel 2148  {cab 2163  {crab 2459   ∩ cin 3128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-rab 2464  df-in 3135

This theorem is referenced by:  nfin  3341  rabbi2dva  3343  ssfidc  6930  suprzubdc  11944  nninfdcex  11945  nnmindc  12026  nnminle  12027  znnen  12390  bj-inex  14510