Theorem inidm 3373

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 3357	1 ⊢ (𝐴 ∩ 𝐴) = 𝐴

Syntax hints:   = wceq 1364   ∈ wcel 2167   ∩ cin 3156

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163

This theorem is referenced by:  inindi  3381  inindir  3382  uneqin  3415  ssdifeq0  3534  intsng  3909  xpindi  4802  xpindir  4803  resindm  4989  ofres  6154  offval2  6155  ofrfval2  6156  suppssof1  6157  ofco  6158  offveqb  6159  ofc1g  6161  ofc2g  6162  caofref  6164  caofrss  6171  caoftrn  6172  undifdc  6994  ofnegsub  9008  ressbasid  12775  strressid  12776  ressinbasd  12779  grpressid  13265  gsumfzmptfidmadd  13547  lcomf  13961  crng2idl  14165  psrbaglesuppg  14306  psraddcl  14314  mplsubgfilemcl  14333  baspartn  14394  epttop  14434  dvaddxxbr  15045  dvmulxxbr  15046  dvaddxx  15047  dvmulxx  15048  dviaddf  15049  dvimulf  15050  plyaddlem1  15091  plyaddlem  15093