Theorem inidm 3285

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 393	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
2	1	ineqri 3269	1 ⊢ (𝐴 ∩ 𝐴) = 𝐴

Syntax hints:   = wceq 1331   ∈ wcel 1480   ∩ cin 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077

This theorem is referenced by:  inindi  3293  inindir  3294  uneqin  3327  ssdifeq0  3445  intsng  3805  xpindi  4674  xpindir  4675  resindm  4861  ofres  5996  offval2  5997  ofrfval2  5998  suppssof1  5999  ofco  6000  offveqb  6001  caofref  6003  caofrss  6006  caoftrn  6007  undifdc  6812  baspartn  12217  epttop  12259  dvaddxxbr  12834  dvmulxxbr  12835  dvaddxx  12836  dvmulxx  12837  dviaddf  12838  dvimulf  12839