Theorem difin 3372

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 615	. . . . . . . 8 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⊥))
2	1	expd 258	. . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → ⊥)))
3		dfnot 1371	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐵 → ⊥))
4	2, 3	syl6ibr 162	. . . . . 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 445	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
7		simpr 110	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	con3i 632	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐵 → ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
9	8	anim2i 342	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
10	6, 9	impbii 126	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
11		eldif 3138	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)))
12		elin 3318	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
13	12	notbii 668	. . . . 5 ⊢ (¬ 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
14	13	anbi2i 457	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
15	11, 14	bitri 184	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
16		eldif 3138	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
17	10, 15, 16	3bitr4i 212	. 2 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∩ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
18	17	eqriv 2174	1 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1353  ⊥wfal 1358   ∈ wcel 2148   ∖ cdif 3126   ∩ cin 3128

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135

This theorem is referenced by:  inssddif  3376  symdif1  3400  notrab  3412  disjdif2  3501  unfiin  6919  bj-charfundcALT  14217