Theorem dfdif2 3868

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 3862	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
2		df-rab 3080	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
3	1, 2	eqtr4i 2785	1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {cab 2736  {crab 3075   ∖ cdif 3856

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 1912  ax-6 1971  ax-7 2016  ax-9 2122  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-cleq 2751  df-rab 3080  df-dif 3862

This theorem is referenced by:  dfdif3  4021  difeq1  4022  difeq2  4023  nfdif  4032  difid  4270  ordintdif  6219  kmlem3  9605  incexc2  15234  cnambfre  35378  alephiso3  40624  sqrtcvallem1  40697