Theorem inidm 3440

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 400	. 2 |- ( ( x e. A /\ x e. A ) <-> x e. A )
2	1	ineqri 3424	1 |- ( A i^i A ) = A

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209   i^i cin 3219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is used by:  inindi  3448  inindir  3449  uneqin  3482  ssdifeq0  3610  intsng  4004  xpindi  4915  xpindir  4916  resindm  5105  ofres  6317  offval2  6318  ofrfval2  6319  suppssof1  6320  ofco  6321  offveqb  6322  ofc1g  6324  ofc2g  6325  caofref  6327  caofrss  6334  caoftrn  6335  suppofss1dcl  6504  suppofss2dcl  6505  undifdc  7231  ofnegsub  9292  ressbasid  13424  strressid  13425  ressinbasd  13428  grpressid  13866  lcomf  14664  crng2idl  14868  psrbaglesuppg  15057  psrbagaddclfi  15061  psrbagcon  15062  psrbagconf1o  15064  psraddcl  15071  mplsubgfilemcl  15090  baspartn  15151  epttop  15191  dvaddxxbr  15802  dvmulxxbr  15803  dvaddxx  15804  dvmulxx  15805  dviaddf  15806  dvimulf  15807  plyaddlem1  15848  plyaddlem  15850