Theorem intprg 4903

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4902. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)

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		preq1 4663	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
2	1	inteqd 4874	. . 3 ⊢ (𝑥 = 𝐴 → ∩ {𝑥, 𝑦} = ∩ {𝐴, 𝑦})
3		ineq1 4181	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
4	2, 3	eqeq12d 2837	. 2 ⊢ (𝑥 = 𝐴 → (∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦) ↔ ∩ {𝐴, 𝑦} = (𝐴 ∩ 𝑦)))
5		preq2 4664	. . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
6	5	inteqd 4874	. . 3 ⊢ (𝑦 = 𝐵 → ∩ {𝐴, 𝑦} = ∩ {𝐴, 𝐵})
7		ineq2 4183	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
8	6, 7	eqeq12d 2837	. 2 ⊢ (𝑦 = 𝐵 → (∩ {𝐴, 𝑦} = (𝐴 ∩ 𝑦) ↔ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)))
9		vex 3498	. . 3 ⊢ 𝑥 ∈ V
10		vex 3498	. . 3 ⊢ 𝑦 ∈ V
11	9, 10	intpr 4902	. 2 ⊢ ∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦)
12	4, 8, 11	vtocl2g 3572	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ∩ cin 3935  {cpr 4563  ∩ 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-3an 1085  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-ral 3143  df-rab 3147  df-v 3497  df-un 3941  df-in 3943  df-sn 4562  df-pr 4564  df-int 4870

This theorem is referenced by:  intsng  4904  inelfi  8876  mreincl  16864  subrgin  19552  lssincl  19731  incld  21645  difelsiga  31387  inelpisys  31408  bj-prmoore  34401  inidl  35302