Theorem inidm 3382

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 3366	1 |- ( A i^i A ) = A

Syntax hints:   = wceq 1373   e. wcel 2176   i^i cin 3165

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  inindi  3390  inindir  3391  uneqin  3424  ssdifeq0  3543  intsng  3919  xpindi  4814  xpindir  4815  resindm  5002  ofres  6175  offval2  6176  ofrfval2  6177  suppssof1  6178  ofco  6179  offveqb  6180  ofc1g  6182  ofc2g  6183  caofref  6185  caofrss  6192  caoftrn  6193  undifdc  7023  ofnegsub  9037  ressbasid  12935  strressid  12936  ressinbasd  12939  grpressid  13426  gsumfzmptfidmadd  13708  lcomf  14122  crng2idl  14326  psrbaglesuppg  14467  psraddcl  14475  mplsubgfilemcl  14494  baspartn  14555  epttop  14595  dvaddxxbr  15206  dvmulxxbr  15207  dvaddxx  15208  dvmulxx  15209  dviaddf  15210  dvimulf  15211  plyaddlem1  15252  plyaddlem  15254