Theorem inidm 3146

Description: Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
inidm	|- ( A i^i A ) = A

Proof of Theorem inidm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anidm 376	. 2 |- ( ( x e. A /\ x e. A ) <-> x e. A )
2	1	ineqri 3130	1 |- ( A i^i A ) = A

Syntax hints:   = wceq 1243   e. wcel 1393   i^i cin 2916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924

This theorem is referenced by:  inindi  3154  inindir  3155  uneqin  3188  ssdifeq0  3305  intsng  3649  xpindi  4471  xpindir  4472  ofres  5725  offval2  5726  ofrfval2  5727  suppssof1  5728  ofco  5729  offveqb  5730  caofref  5732  caofrss  5735  caoftrn  5736