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 ∩ 𝐴) = 𝐴

Proof of Theorem inidm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anidm 400	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
2	1	ineqri 3424	1 ⊢ (𝐴 ∩ 𝐴) = 𝐴

Syntax hints:   = wceq 1402   ∈ wcel 2209   ∩ cin 3219

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 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 theorem 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 referenced by:  inindi  3448  inindir  3449  uneqin  3482  ssdifeq0  3607  intsng  3999  xpindi  4910  xpindir  4911  resindm  5100  ofres  6307  offval2  6308  ofrfval2  6309  suppssof1  6310  ofco  6311  offveqb  6312  ofc1g  6314  ofc2g  6315  caofref  6317  caofrss  6324  caoftrn  6325  suppofss1dcl  6494  suppofss2dcl  6495  undifdc  7221  ofnegsub  9282  ressbasid  13401  strressid  13402  ressinbasd  13405  grpressid  13843  lcomf  14636  crng2idl  14840  psrbaglesuppg  14980  psrbagaddclfi  14984  psrbagcon  14985  psrbagconf1o  14987  psraddcl  14994  mplsubgfilemcl  15013  baspartn  15074  epttop  15114  dvaddxxbr  15725  dvmulxxbr  15726  dvaddxx  15727  dvmulxx  15728  dviaddf  15729  dvimulf  15730  plyaddlem1  15771  plyaddlem  15773