Theorem indif 3365

Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
indif	⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Proof of Theorem indif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anabs5 563	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		elin 3305	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)))
3		eldif 3125	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
4	3	anbi2i 453	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
5	2, 4	bitri 183	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
6	1, 5, 3	3bitr4i 211	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐴 ∖ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
7	6	eqriv 2162	1 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1343   ∈ wcel 2136   ∖ cdif 3113   ∩ cin 3115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122

This theorem is referenced by:  resdif  5454