Theorem inidm 3344

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 3328	1 ⊢ (𝐴 ∩ 𝐴) = 𝐴

Syntax hints:   = wceq 1353   ∈ wcel 2148   ∩ cin 3128

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 2739  df-in 3135

This theorem is referenced by:  inindi  3352  inindir  3353  uneqin  3386  ssdifeq0  3505  intsng  3878  xpindi  4762  xpindir  4763  resindm  4949  ofres  6096  offval2  6097  ofrfval2  6098  suppssof1  6099  ofco  6100  offveqb  6101  caofref  6103  caofrss  6106  caoftrn  6107  undifdc  6922  strressid  12529  ressinbasd  12532  grpressid  12930  baspartn  13520  epttop  13560  dvaddxxbr  14135  dvmulxxbr  14136  dvaddxx  14137  dvmulxx  14138  dviaddf  14139  dvimulf  14140