Theorem dfdif2 3899

Description: Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)

Assertion

Ref	Expression
dfdif2	⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfdif2

Step	Hyp	Ref	Expression
1		df-dif 3893	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
2		df-rab 3391	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
3	1, 2	eqtr4i 2763	1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  {crab 3390   ∖ cdif 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-rab 3391  df-dif 3893

This theorem is referenced by:  dfdif3  4058  dfdif3OLD  4059  difeq1  4060  difeq2  4061  nfdifOLD  4071  difid  4317  ordintdif  6369  kmlem3  10069  incexc2  15797  cnambfre  38006  alephiso3  44007  sqrtcvallem1  44079