Theorem intpr 1990

Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42.

Hypotheses

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

Assertion

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

Proof of Theorem intpr

Step	Hyp	Ref	Expression
1		19.26 749	. . . 4 ⊢ (∀y((y = A → x ∈ y) ∧ (y = B → x ∈ y)) ↔ (∀y(y = A → x ∈ y) ∧ ∀y(y = B → x ∈ y)))
2		visset 1350	. . . . . . . 8 ⊢ y ∈ V
3	2	elpr 1823	. . . . . . 7 ⊢ (y ∈ {A, B} ↔ (y = A ∨ y = B))
4	3	imbi1i 161	. . . . . 6 ⊢ ((y ∈ {A, B} → x ∈ y) ↔ ((y = A ∨ y = B) → x ∈ y))
5		jaob 328	. . . . . 6 ⊢ (((y = A ∨ y = B) → x ∈ y) ↔ ((y = A → x ∈ y) ∧ (y = B → x ∈ y)))
6	4, 5	bitr 151	. . . . 5 ⊢ ((y ∈ {A, B} → x ∈ y) ↔ ((y = A → x ∈ y) ∧ (y = B → x ∈ y)))
7	6	bial 695	. . . 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 1376	. . . . 5 ⊢ (x ∈ A ↔ ∀y(y = A → x ∈ y))
10		intpr.2	. . . . . 6 ⊢ B ∈ V
11	10	clel4 1376	. . . . 5 ⊢ (x ∈ B ↔ ∀y(y = B → x ∈ y))
12	9, 11	anbi12i 369	. . . 4 ⊢ ((x ∈ A ∧ x ∈ B) ↔ (∀y(y = A → x ∈ y) ∧ ∀y(y = B → x ∈ y)))
13	1, 7, 12	3bitr4 158	. . 3 ⊢ (∀y(y ∈ {A, B} → x ∈ y) ↔ (x ∈ A ∧ x ∈ B))
14		visset 1350	. . . 4 ⊢ x ∈ V
15	14	elint 1971	. . 3 ⊢ (x ∈ ∩{A, B} ↔ ∀y(y ∈ {A, B} → x ∈ y))
16		elin 1635	. . 3 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
17	13, 15, 16	3bitr4 158	. 2 ⊢ (x ∈ ∩{A, B} ↔ x ∈ (A ∩ B))
18	17	cleqri 1101	1 ⊢ ∩{A, B} = (A ∩ B)

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196  ∀wal 672   ∈ wel 803   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∩ cin 1486  {cpr 1809  ∩cint 1965

This theorem is referenced by:  intsn 1991  op1stb 1992  fiint 3445  shincl 5332  chincl 5382

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-sn 1811  df-pr 1812  df-int 1966

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