Theorem inidm 3290

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 394	. 2 |- ( ( x e. A /\ x e. A ) <-> x e. A )
2	1	ineqri 3274	1 |- ( A i^i A ) = A

Syntax hints:   = wceq 1332   e. wcel 1481   i^i cin 3075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082

This theorem is referenced by:  inindi  3298  inindir  3299  uneqin  3332  ssdifeq0  3450  intsng  3813  xpindi  4682  xpindir  4683  resindm  4869  ofres  6004  offval2  6005  ofrfval2  6006  suppssof1  6007  ofco  6008  offveqb  6009  caofref  6011  caofrss  6014  caoftrn  6015  undifdc  6820  baspartn  12256  epttop  12298  dvaddxxbr  12873  dvmulxxbr  12874  dvaddxx  12875  dvmulxx  12876  dviaddf  12877  dvimulf  12878