Theorem inidm 3326

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 394	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
2	1	ineqri 3310	1 ⊢ (𝐴 ∩ 𝐴) = 𝐴

Syntax hints:   = wceq 1342   ∈ wcel 2135   ∩ cin 3110

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117

This theorem is referenced by:  inindi  3334  inindir  3335  uneqin  3368  ssdifeq0  3486  intsng  3852  xpindi  4733  xpindir  4734  resindm  4920  ofres  6058  offval2  6059  ofrfval2  6060  suppssof1  6061  ofco  6062  offveqb  6063  caofref  6065  caofrss  6068  caoftrn  6069  undifdc  6880  baspartn  12589  epttop  12631  dvaddxxbr  13206  dvmulxxbr  13207  dvaddxx  13208  dvmulxx  13209  dviaddf  13210  dvimulf  13211