Theorem intun 3855

Description: The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)

Assertion

Ref	Expression
intun	⊢ ∩ (𝐴 ∪ 𝐵) = (∩ 𝐴 ∩ ∩ 𝐵)

Proof of Theorem intun

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.26 1469	. . . 4 ⊢ (∀𝑦((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)) ↔ (∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
2		elun 3263	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵))
3	2	imbi1i 237	. . . . . 6 ⊢ ((𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝑦))
4		jaob 700	. . . . . 6 ⊢ (((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
5	3, 4	bitri 183	. . . . 5 ⊢ ((𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
6	5	albii 1458	. . . 4 ⊢ (∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ ∀𝑦((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
7		vex 2729	. . . . . 6 ⊢ 𝑥 ∈ V
8	7	elint 3830	. . . . 5 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))
9	7	elint 3830	. . . . 5 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦))
10	8, 9	anbi12i 456	. . . 4 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵) ↔ (∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
11	1, 6, 10	3bitr4i 211	. . 3 ⊢ (∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ (𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵))
12	7	elint 3830	. . 3 ⊢ (𝑥 ∈ ∩ (𝐴 ∪ 𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦))
13		elin 3305	. . 3 ⊢ (𝑥 ∈ (∩ 𝐴 ∩ ∩ 𝐵) ↔ (𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵))
14	11, 12, 13	3bitr4i 211	. 2 ⊢ (𝑥 ∈ ∩ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (∩ 𝐴 ∩ ∩ 𝐵))
15	14	eqriv 2162	1 ⊢ ∩ (𝐴 ∪ 𝐵) = (∩ 𝐴 ∩ ∩ 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698  ∀wal 1341   = wceq 1343   ∈ wcel 2136   ∪ cun 3114   ∩ cin 3115  ∩ cint 3824

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-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-un 3120  df-in 3122  df-int 3825

This theorem is referenced by:  intunsn  3862  riinint  4865