Theorem inidm 3432

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 3416	1 |- ( A i^i A ) = A

Syntax hints:   = wceq 1398   e. wcel 2205   i^i cin 3212

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219

This theorem is referenced by:  inindi  3440  inindir  3441  uneqin  3474  ssdifeq0  3594  intsng  3985  xpindi  4892  xpindir  4893  resindm  5082  ofres  6283  offval2  6284  ofrfval2  6285  suppssof1  6286  ofco  6287  offveqb  6288  ofc1g  6290  ofc2g  6291  caofref  6293  caofrss  6300  caoftrn  6301  suppofss1dcl  6466  suppofss2dcl  6467  undifdc  7186  ofnegsub  9238  ressbasid  13300  strressid  13301  ressinbasd  13304  grpressid  13791  gsumfzmptfidmadd  14073  lcomf  14492  crng2idl  14696  psrbaglesuppg  14838  psrbagaddclfi  14842  psrbagcon  14843  psrbagconf1o  14845  psraddcl  14852  mplsubgfilemcl  14871  baspartn  14932  epttop  14972  dvaddxxbr  15583  dvmulxxbr  15584  dvaddxx  15585  dvmulxx  15586  dviaddf  15587  dvimulf  15588  plyaddlem1  15629  plyaddlem  15631