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Theorem difrab0eq 3990
 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 3896 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ (𝑉 ∖ {𝑥𝑉𝜑}) = ∅)
2 ssrabeq 3651 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
31, 2bitr3i 265 1 ((𝑉 ∖ {𝑥𝑉𝜑}) = ∅ ↔ 𝑉 = {𝑥𝑉𝜑})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  frgraregorufr0  26579  frgrregorufr0  41489
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