Theorem inidm 1649

Description: Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26.

Assertion

Ref	Expression
inidm	⊢ (A ∩ A) = A

Proof of Theorem inidm

Step	Hyp	Ref	Expression
1		anidm 331	. 2 ⊢ ((x ∈ A ∧ x ∈ A) ↔ x ∈ A)
2	1	ineqri 1637	1 ⊢ (A ∩ A) = A

Syntax hints:   = wceq 1091   ∈ wcel 1092   ∩ cin 1486

This theorem is referenced by:  inindi 1654  inindir 1655  ssin 1659  intsn 1991  xpindi 2497  xpindir 2498  resabs1 2592  chjidmt 5436

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-v 1349  df-in 1491

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