Theorem inidm 3434

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 3418	1 ⊢ (𝐴 ∩ 𝐴) = 𝐴

Syntax hints:   = wceq 1398   ∈ wcel 2205   ∩ cin 3213

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220

This theorem is referenced by:  inindi  3442  inindir  3443  uneqin  3476  ssdifeq0  3596  intsng  3988  xpindi  4895  xpindir  4896  resindm  5085  ofres  6290  offval2  6291  ofrfval2  6292  suppssof1  6293  ofco  6294  offveqb  6295  ofc1g  6297  ofc2g  6298  caofref  6300  caofrss  6307  caoftrn  6308  suppofss1dcl  6477  suppofss2dcl  6478  undifdc  7197  ofnegsub  9253  ressbasid  13367  strressid  13368  ressinbasd  13371  grpressid  13816  gsumfzmptfidmadd  14092  lcomf  14601  crng2idl  14805  psrbaglesuppg  14947  psrbagaddclfi  14951  psrbagcon  14952  psrbagconf1o  14954  psraddcl  14961  mplsubgfilemcl  14980  baspartn  15041  epttop  15081  dvaddxxbr  15692  dvmulxxbr  15693  dvaddxx  15694  dvmulxx  15695  dviaddf  15696  dvimulf  15697  plyaddlem1  15738  plyaddlem  15740