Theorem difin 3318

Description: Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem difin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-in2 605	. . . . . . . 8 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⊥))
2	1	expd 256	. . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → ⊥)))
3		dfnot 1350	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐵 → ⊥))
4	2, 3	syl6ibr 161	. . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
5	4	com12 30	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ 𝐵))
6	5	imdistani 442	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
7		simpr 109	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	con3i 622	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐵 → ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
9	8	anim2i 340	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
10	6, 9	impbii 125	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
11		eldif 3085	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)))
12		elin 3264	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
13	12	notbii 658	. . . . 5 ⊢ (¬ 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
14	13	anbi2i 453	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
15	11, 14	bitri 183	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
16		eldif 3085	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
17	10, 15, 16	3bitr4i 211	. 2 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∩ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
18	17	eqriv 2137	1 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1332  ⊥wfal 1337   ∈ wcel 1481   ∖ cdif 3073   ∩ cin 3075

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-in 3082

This theorem is referenced by:  inssddif  3322  symdif1  3346  notrab  3358  disjdif2  3446  unfiin  6822