Theorem intun 4901

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 1867	. . . 4 ⊢ (∀𝑦((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)) ↔ (∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
2		elunant 4154	. . . . 5 ⊢ ((𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
3	2	albii 1816	. . . 4 ⊢ (∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ ∀𝑦((𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)))
4		vex 3498	. . . . . 6 ⊢ 𝑥 ∈ V
5	4	elint 4875	. . . . 5 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))
6	4	elint 4875	. . . . 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 305	. . 3 ⊢ (∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦) ↔ (𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵))
9	4	elint 4875	. . 3 ⊢ (𝑥 ∈ ∩ (𝐴 ∪ 𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝑦))
10		elin 4169	. . 3 ⊢ (𝑥 ∈ (∩ 𝐴 ∩ ∩ 𝐵) ↔ (𝑥 ∈ ∩ 𝐴 ∧ 𝑥 ∈ ∩ 𝐵))
11	8, 9, 10	3bitr4i 305	. 2 ⊢ (𝑥 ∈ ∩ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (∩ 𝐴 ∩ ∩ 𝐵))
12	11	eqriv 2818	1 ⊢ ∩ (𝐴 ∪ 𝐵) = (∩ 𝐴 ∩ ∩ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1531   = wceq 1533   ∈ wcel 2110   ∪ cun 3934   ∩ cin 3935  ∩ cint 4869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-un 3941  df-in 3943  df-int 4870

This theorem is referenced by:  intunsn  4908  riinint  5834  fiin  8880  elfiun  8888  elrfi  39284