Theorem indif 3406

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 573	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		elin 3346	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)))
3		eldif 3166	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
4	3	anbi2i 457	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
5	2, 4	bitri 184	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
6	1, 5, 3	3bitr4i 212	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐴 ∖ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
7	6	eqriv 2193	1 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1364   ∈ wcel 2167   ∖ cdif 3154   ∩ cin 3156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163

This theorem is referenced by:  resdif  5526