Theorem difin 3174

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 545	. . . . . . . 8 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⊥))
2	1	expd 245	. . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → ⊥)))
3		dfnot 1262	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐵 → ⊥))
4	2, 3	syl6ibr 151	. . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
5	4	com12 27	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ 𝐵))
6	5	imdistani 419	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
7		simpr 103	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	con3i 562	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐵 → ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
9	8	anim2i 324	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
10	6, 9	impbii 117	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
11		eldif 2927	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)))
12		elin 3126	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
13	12	notbii 594	. . . . 5 ⊢ (¬ 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
14	13	anbi2i 430	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
15	11, 14	bitri 173	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
16		eldif 2927	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
17	10, 15, 16	3bitr4i 201	. 2 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∩ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
18	17	eqriv 2037	1 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1243  ⊥wfal 1248   ∈ wcel 1393   ∖ cdif 2914   ∩ cin 2916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924

This theorem is referenced by:  inssddif  3178  symdif1  3202  notrab  3214