Theorem intiin 2027

Description: Class intersection in terms of indexed intersection. Definition of [Stoll] p. 44.

Assertion

Ref	Expression
intiin	⊢ ∩A = ∩x ∈ A x

Distinct variable group(s):   x,A

Proof of Theorem intiin

Step	Hyp	Ref	Expression
1		dfint2 1967	. 2 ⊢ ∩A = {y∣∀x ∈ A y ∈ x}
2		df-iin 1997	. 2 ⊢ ∩x ∈ A x = {y∣∀x ∈ A y ∈ x}
3	1, 2	eqtr4 1122	1 ⊢ ∩A = ∩x ∈ A x

Syntax hints:   ∈ wel 803  {cab 1090   = wceq 1091  ∀wral 1201  ∩cint 1965  ∩ciin 1995

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-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-int 1966  df-iin 1997

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