Theorem inidm 3413

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 396	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
2	1	ineqri 3397	1 ⊢ (𝐴 ∩ 𝐴) = 𝐴

Syntax hints:   = wceq 1395   ∈ wcel 2200   ∩ cin 3196

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203

This theorem is referenced by:  inindi  3421  inindir  3422  uneqin  3455  ssdifeq0  3574  intsng  3957  xpindi  4857  xpindir  4858  resindm  5047  ofres  6233  offval2  6234  ofrfval2  6235  suppssof1  6236  ofco  6237  offveqb  6238  ofc1g  6240  ofc2g  6241  caofref  6243  caofrss  6250  caoftrn  6251  undifdc  7086  ofnegsub  9109  ressbasid  13103  strressid  13104  ressinbasd  13107  grpressid  13594  gsumfzmptfidmadd  13876  lcomf  14291  crng2idl  14495  psrbaglesuppg  14636  psraddcl  14644  mplsubgfilemcl  14663  baspartn  14724  epttop  14764  dvaddxxbr  15375  dvmulxxbr  15376  dvaddxx  15377  dvmulxx  15378  dviaddf  15379  dvimulf  15380  plyaddlem1  15421  plyaddlem  15423