Theorem intpr 3959

Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)

Hypotheses

Ref	Expression
intpr.1	⊢ A ∈ V
intpr.2	⊢ B ∈ V

Assertion

Ref	Expression
intpr	⊢ ∩{A, B} = (A ∩ B)

Proof of Theorem intpr

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

Step	Hyp	Ref	Expression
1		19.26 1593	. . . 4 ⊢ (∀y((y = A → x ∈ y) ∧ (y = B → x ∈ y)) ↔ (∀y(y = A → x ∈ y) ∧ ∀y(y = B → x ∈ y)))
2		vex 2862	. . . . . . . 8 ⊢ y ∈ V
3	2	elpr 3751	. . . . . . 7 ⊢ (y ∈ {A, B} ↔ (y = A ∨ y = B))
4	3	imbi1i 315	. . . . . 6 ⊢ ((y ∈ {A, B} → x ∈ y) ↔ ((y = A ∨ y = B) → x ∈ y))
5		jaob 758	. . . . . 6 ⊢ (((y = A ∨ y = B) → x ∈ y) ↔ ((y = A → x ∈ y) ∧ (y = B → x ∈ y)))
6	4, 5	bitri 240	. . . . 5 ⊢ ((y ∈ {A, B} → x ∈ y) ↔ ((y = A → x ∈ y) ∧ (y = B → x ∈ y)))
7	6	albii 1566	. . . 4 ⊢ (∀y(y ∈ {A, B} → x ∈ y) ↔ ∀y((y = A → x ∈ y) ∧ (y = B → x ∈ y)))
8		intpr.1	. . . . . 6 ⊢ A ∈ V
9	8	clel4 2978	. . . . 5 ⊢ (x ∈ A ↔ ∀y(y = A → x ∈ y))
10		intpr.2	. . . . . 6 ⊢ B ∈ V
11	10	clel4 2978	. . . . 5 ⊢ (x ∈ B ↔ ∀y(y = B → x ∈ y))
12	9, 11	anbi12i 678	. . . 4 ⊢ ((x ∈ A ∧ x ∈ B) ↔ (∀y(y = A → x ∈ y) ∧ ∀y(y = B → x ∈ y)))
13	1, 7, 12	3bitr4i 268	. . 3 ⊢ (∀y(y ∈ {A, B} → x ∈ y) ↔ (x ∈ A ∧ x ∈ B))
14		vex 2862	. . . 4 ⊢ x ∈ V
15	14	elint 3932	. . 3 ⊢ (x ∈ ∩{A, B} ↔ ∀y(y ∈ {A, B} → x ∈ y))
16		elin 3219	. . 3 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
17	13, 15, 16	3bitr4i 268	. 2 ⊢ (x ∈ ∩{A, B} ↔ x ∈ (A ∩ B))
18	17	eqriv 2350	1 ⊢ ∩{A, B} = (A ∩ B)

Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208  {cpr 3738  ∩cint 3926

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-sn 3741  df-pr 3742  df-int 3927

This theorem is referenced by:  intprg  3960  uniintsn  3963