Theorem intprg 3961

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3960. 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 3800	. . . 4 ⊢ (x = A → {x, y} = {A, y})
2	1	inteqd 3932	. . 3 ⊢ (x = A → ∩{x, y} = ∩{A, y})
3		ineq1 3451	. . 3 ⊢ (x = A → (x ∩ y) = (A ∩ y))
4	2, 3	eqeq12d 2367	. 2 ⊢ (x = A → (∩{x, y} = (x ∩ y) ↔ ∩{A, y} = (A ∩ y)))
5		preq2 3801	. . . 4 ⊢ (y = B → {A, y} = {A, B})
6	5	inteqd 3932	. . 3 ⊢ (y = B → ∩{A, y} = ∩{A, B})
7		ineq2 3452	. . 3 ⊢ (y = B → (A ∩ y) = (A ∩ B))
8	6, 7	eqeq12d 2367	. 2 ⊢ (y = B → (∩{A, y} = (A ∩ y) ↔ ∩{A, B} = (A ∩ B)))
9		vex 2863	. . 3 ⊢ x ∈ V
10		vex 2863	. . 3 ⊢ y ∈ V
11	9, 10	intpr 3960	. 2 ⊢ ∩{x, y} = (x ∩ y)
12	4, 8, 11	vtocl2g 2919	1 ⊢ ((A ∈ V ∧ B ∈ W) → ∩{A, B} = (A ∩ B))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3209  {cpr 3739  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-sn 3742  df-pr 3743  df-int 3928

This theorem is referenced by:  intsng  3962