Theorem dfdif2 3908

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 3902	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
2		df-rab 3398	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
3	1, 2	eqtr4i 2760	1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712  {crab 3397   ∖ cdif 3896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2726  df-rab 3398  df-dif 3902

This theorem is referenced by:  dfdif3  4067  dfdif3OLD  4068  difeq1  4069  difeq2  4070  nfdifOLD  4080  difid  4326  ordintdif  6366  kmlem3  10061  incexc2  15759  cnambfre  37808  alephiso3  43742  sqrtcvallem1  43814