Theorem intprg 4945

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4946. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.) (Proof shortened by BJ, 1-Sep-2024.)

Assertion

Ref	Expression
intprg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))

Proof of Theorem intprg

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

Step	Hyp	Ref	Expression
1		vex 3457	. . . . 5 ⊢ 𝑥 ∈ V
2	1	elint 4917	. . . 4 ⊢ (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦))
3		vex 3457	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	3	elpr 4613	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
5	4	imbi1i 352	. . . . . 6 ⊢ ((𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦))
6		jaob 976	. . . . . 6 ⊢ (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
7	5, 6	bitri 278	. . . . 5 ⊢ ((𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
8	7	albii 1847	. . . 4 ⊢ (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ∀𝑦((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
9		19.26 1898	. . . 4 ⊢ (∀𝑦((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)) ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
10	2, 8, 9	3bitri 300	. . 3 ⊢ (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
11		elin 3920	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
12		clel4g 3621	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↔ ∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦)))
13		clel4g 3621	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐵 ↔ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
14	12, 13	bi2anan9 649	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦))))
15	11, 14	bitr2id 287	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦)) ↔ 𝑥 ∈ (𝐴 ∩ 𝐵)))
16	10, 15	bitrid 286	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴 ∩ 𝐵)))
17	16	eqrdv 2759	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860  ∀wal 1566   = wceq 1568   ∈ wcel 2141   ∩ cin 3903  {cpr 4590  ∩ cint 4911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-in 3911  df-sn 4589  df-pr 4591  df-int 4912

This theorem is referenced by:  intpr  4946  intsng  4947  inelfi  9377  mreincl  17650  subrngin  20645  subrgin  20680  lssincl  21065  incld  23179  inlidl  33695  difelsiga  34489  inelpisys  34510  bj-prmoore  37701  inidl  38625  toplatmeet  49726