Theorem inidm 3346

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 3330	1 |- ( A i^i A ) = A

Syntax hints:   = wceq 1353   e. wcel 2148   i^i cin 3130

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137

This theorem is referenced by:  inindi  3354  inindir  3355  uneqin  3388  ssdifeq0  3507  intsng  3880  xpindi  4764  xpindir  4765  resindm  4951  ofres  6099  offval2  6100  ofrfval2  6101  suppssof1  6102  ofco  6103  offveqb  6104  caofref  6106  caofrss  6109  caoftrn  6110  undifdc  6925  strressid  12532  ressinbasd  12535  grpressid  12936  lcomf  13422  baspartn  13635  epttop  13675  dvaddxxbr  14250  dvmulxxbr  14251  dvaddxx  14252  dvmulxx  14253  dviaddf  14254  dvimulf  14255