Theorem intiin 4021

Description: Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)

Assertion

Ref	Expression
intiin	⊢ ∩A = ∩x ∈ A x

Distinct variable group:   x,A

Proof of Theorem intiin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfint2 3929	. 2 ⊢ ∩A = {y ∣ ∀x ∈ A y ∈ x}
2		df-iin 3973	. 2 ⊢ ∩x ∈ A x = {y ∣ ∀x ∈ A y ∈ x}
3	1, 2	eqtr4i 2376	1 ⊢ ∩A = ∩x ∈ A x

Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∩cint 3927  ∩ciin 3971

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-ral 2620  df-int 3928  df-iin 3973

This theorem is referenced by: (None)