Theorem inidm 3345

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 3329	1 ⊢ (𝐴 ∩ 𝐴) = 𝐴

Syntax hints:   = wceq 1353   ∈ wcel 2148   ∩ cin 3129

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136

This theorem is referenced by:  inindi  3353  inindir  3354  uneqin  3387  ssdifeq0  3506  intsng  3879  xpindi  4763  xpindir  4764  resindm  4950  ofres  6097  offval2  6098  ofrfval2  6099  suppssof1  6100  ofco  6101  offveqb  6102  caofref  6104  caofrss  6107  caoftrn  6108  undifdc  6923  strressid  12530  ressinbasd  12533  grpressid  12931  lcomf  13417  baspartn  13553  epttop  13593  dvaddxxbr  14168  dvmulxxbr  14169  dvaddxx  14170  dvmulxx  14171  dviaddf  14172  dvimulf  14173