MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difrab0eq Structured version   Visualization version   GIF version

Theorem difrab0eq 4427
Description: If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Assertion
Ref Expression
difrab0eq ((𝑉 ∖ {𝑥𝑉𝜑}) = ∅ ↔ 𝑉 = {𝑥𝑉𝜑})
Distinct variable group:   𝑥,𝑉
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem difrab0eq
StepHypRef Expression
1 ssdif0 4321 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ (𝑉 ∖ {𝑥𝑉𝜑}) = ∅)
2 ssrabeq 4040 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
31, 2bitr3i 276 1 ((𝑉 ∖ {𝑥𝑉𝜑}) = ∅ ↔ 𝑉 = {𝑥𝑉𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  {crab 3405  cdif 3905  wss 3908  c0 4280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4281
This theorem is referenced by:  frgrregorufr0  29154
  Copyright terms: Public domain W3C validator