Theorem dfss 3936

Description: Variant of subclass definition dfss2 3935. (Contributed by NM, 21-Jun-1993.)

Assertion

Ref	Expression
dfss	⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))

Proof of Theorem dfss

Step	Hyp	Ref	Expression
1		dfss2 3935	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		eqcom 2737	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 ↔ 𝐴 = (𝐴 ∩ 𝐵))
3	1, 2	bitri 275	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))

Syntax hints:   ↔ wb 206   = wceq 1540   ∩ cin 3916   ⊆ wss 3917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-in 3924  df-ss 3934

This theorem is referenced by:  iinrab2  5037  wefrc  5635  cnvcnv  6168  ordtri2or3  6437  onelini  6455  funimass1  6601  sbthlem5  9061  dmaddpi  10850  dmmulpi  10851  smndex1bas  18840  restcldi  23067  cmpsublem  23293  ustuqtop5  24140  tgioo  24691  cphsscph  25158  mdbr3  32233  mdbr4  32234  ssmd1  32247  xrge00  32960  esumpfinvallem  34071  measxun2  34207  eulerpartgbij  34370  reprfz1  34622  bj-ismooredr2  37105  bndss  37787  redundss3  38626  dfrcl2  43670  isotone2  44045  wfac8prim  44999  restuni4  45122  fourierdlem93  46204  sge0resplit  46411  mbfresmf  46744