Theorem dfdif2 3896

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 3890	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
2		df-rab 3073	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
3	1, 2	eqtr4i 2769	1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  {crab 3068   ∖ cdif 3884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-rab 3073  df-dif 3890

This theorem is referenced by:  dfdif3  4049  difeq1  4050  difeq2  4051  nfdif  4060  difid  4304  ordintdif  6315  kmlem3  9908  incexc2  15550  cnambfre  35825  alephiso3  41166  sqrtcvallem1  41239