Theorem intun 4935

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 1871	. . . 4 ⊢ (∀𝑦((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)) ↔ (∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
2		elunant 4136	. . . . 5 ⊢ ((𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
3	2	albii 1820	. . . 4 ⊢ (∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ ∀𝑦((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
4		vex 3444	. . . . . 6 ⊢ 𝑥 ∈ V
5	4	elint 4908	. . . . 5 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))
6	4	elint 4908	. . . . 5 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦))
7	5, 6	anbi12i 628	. . . 4 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵) ↔ (∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
8	1, 3, 7	3bitr4i 303	. . 3 ⊢ (∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ (𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵))
9	4	elint 4908	. . 3 ⊢ (𝑥 ∈ ∩ (𝐴 ∪ 𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦))
10		elin 3917	. . 3 ⊢ (𝑥 ∈ (∩ 𝐴 ∩ ∩ 𝐵) ↔ (𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵))
11	8, 9, 10	3bitr4i 303	. 2 ⊢ (𝑥 ∈ ∩ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (∩ 𝐴 ∩ ∩ 𝐵))
12	11	eqriv 2733	1 ⊢ ∩ (𝐴 ∪ 𝐵) = (∩ 𝐴 ∩ ∩ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113   ∪ cun 3899   ∩ cin 3900  ∩ cint 4902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-un 3906  df-in 3908  df-int 4903

This theorem is referenced by:  intunsn  4942  riinint  5921  fiin  9325  elfiun  9333  elrfi  42936