Theorem inidm 3382

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 3366	1 ⊢ (𝐴 ∩ 𝐴) = 𝐴

Syntax hints:   = wceq 1373   ∈ wcel 2176   ∩ cin 3165

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  inindi  3390  inindir  3391  uneqin  3424  ssdifeq0  3543  intsng  3919  xpindi  4813  xpindir  4814  resindm  5001  ofres  6173  offval2  6174  ofrfval2  6175  suppssof1  6176  ofco  6177  offveqb  6178  ofc1g  6180  ofc2g  6181  caofref  6183  caofrss  6190  caoftrn  6191  undifdc  7021  ofnegsub  9035  ressbasid  12902  strressid  12903  ressinbasd  12906  grpressid  13393  gsumfzmptfidmadd  13675  lcomf  14089  crng2idl  14293  psrbaglesuppg  14434  psraddcl  14442  mplsubgfilemcl  14461  baspartn  14522  epttop  14562  dvaddxxbr  15173  dvmulxxbr  15174  dvaddxx  15175  dvmulxx  15176  dviaddf  15177  dvimulf  15178  plyaddlem1  15219  plyaddlem  15221