Theorem inidm 3418

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 3402	1 ⊢ (𝐴 ∩ 𝐴) = 𝐴

Syntax hints:   = wceq 1398   ∈ wcel 2202   ∩ cin 3200

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207

This theorem is referenced by:  inindi  3426  inindir  3427  uneqin  3460  ssdifeq0  3579  intsng  3967  xpindi  4871  xpindir  4872  resindm  5061  ofres  6259  offval2  6260  ofrfval2  6261  suppssof1  6262  ofco  6263  offveqb  6264  ofc1g  6266  ofc2g  6267  caofref  6269  caofrss  6276  caoftrn  6277  suppofss1dcl  6442  suppofss2dcl  6443  undifdc  7159  ofnegsub  9201  ressbasid  13233  strressid  13234  ressinbasd  13237  grpressid  13724  gsumfzmptfidmadd  14006  lcomf  14423  crng2idl  14627  psrbaglesuppg  14768  psrbagcon  14772  psrbagconf1o  14774  psraddcl  14781  mplsubgfilemcl  14800  baspartn  14861  epttop  14901  dvaddxxbr  15512  dvmulxxbr  15513  dvaddxx  15514  dvmulxx  15515  dviaddf  15516  dvimulf  15517  plyaddlem1  15558  plyaddlem  15560