Theorem inidm 3390

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 396	. 2 |- ( ( x e. A /\ x e. A ) <-> x e. A )
2	1	ineqri 3374	1 |- ( A i^i A ) = A

Syntax hints:   = wceq 1373   e. wcel 2178   i^i cin 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180

This theorem is referenced by:  inindi  3398  inindir  3399  uneqin  3432  ssdifeq0  3551  intsng  3933  xpindi  4831  xpindir  4832  resindm  5020  ofres  6196  offval2  6197  ofrfval2  6198  suppssof1  6199  ofco  6200  offveqb  6201  ofc1g  6203  ofc2g  6204  caofref  6206  caofrss  6213  caoftrn  6214  undifdc  7047  ofnegsub  9070  ressbasid  13017  strressid  13018  ressinbasd  13021  grpressid  13508  gsumfzmptfidmadd  13790  lcomf  14204  crng2idl  14408  psrbaglesuppg  14549  psraddcl  14557  mplsubgfilemcl  14576  baspartn  14637  epttop  14677  dvaddxxbr  15288  dvmulxxbr  15289  dvaddxx  15290  dvmulxx  15291  dviaddf  15292  dvimulf  15293  plyaddlem1  15334  plyaddlem  15336