Theorem intsn 1991

Description: The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41.

Hypothesis

Ref	Expression
intsn.1	⊢ A ∈ V

Assertion

Ref	Expression
intsn	⊢ ∩{A} = A

Proof of Theorem intsn

Step	Hyp	Ref	Expression
1		dfsn2 1819	. . 3 ⊢ {A} = {A, A}
2	1	inteqi 1969	. 2 ⊢ ∩{A} = ∩{A, A}
3		intsn.1	. . 3 ⊢ A ∈ V
4	3, 3	intpr 1990	. 2 ⊢ ∩{A, A} = (A ∩ A)
5		inidm 1649	. 2 ⊢ (A ∩ A) = A
6	2, 4, 5	3eqtr 1123	1 ⊢ ∩{A} = A

Syntax hints:   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∩ cin 1486  {csn 1808  {cpr 1809  ∩cint 1965

This theorem is referenced by:  op1stb 1992  intunsn 1993  op2ndb 2638  cf0 3705  cflecard 3707  cfom 3710

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-ral 1205  df-v 1349  df-un 1490  df-in 1491  df-sn 1811  df-pr 1812  df-int 1966

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