Theorem intun 4939

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 1902	. . . 4 ⊢ (∀𝑦((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)) ↔ (∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
2		elunant 4182	. . . . 5 ⊢ ((𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
3	2	albii 1851	. . . 4 ⊢ (∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ ∀𝑦((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
4		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
5	4	elint 4912	. . . . 5 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))
6	4	elint 4912	. . . . 5 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦))
7	5, 6	anbi12i 640	. . . 4 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵) ↔ (∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
8	1, 3, 7	3bitr4i 306	. . 3 ⊢ (∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ (𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵))
9	4	elint 4912	. . 3 ⊢ (𝑥 ∈ ∩ (𝐴 ∪ 𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦))
10		elin 4093	. . 3 ⊢ (𝑥 ∈ (∩ 𝐴 ∩ ∩ 𝐵) ↔ (𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵))
11	8, 9, 10	3bitr4i 306	. 2 ⊢ (𝑥 ∈ ∩ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (∩ 𝐴 ∩ ∩ 𝐵))
12	11	eqriv 2757	1 ⊢ ∩ (𝐴 ∪ 𝐵) = (∩ 𝐴 ∩ ∩ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144   ∩ cin 4089   ∪ cun 4149  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-un 4151  df-int 4907

This theorem is used by:  intunsn  4946  riinint  5953  fiin  9414  elfiun  9422  elrfi  43748