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Theorem difrab0eq 4174
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 4077 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ (𝑉 ∖ {𝑥𝑉𝜑}) = ∅)
2 ssrabeq 3823 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
31, 2bitr3i 266 1 ((𝑉 ∖ {𝑥𝑉𝜑}) = ∅ ↔ 𝑉 = {𝑥𝑉𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1624  {crab 3046  cdif 3704  wss 3707  c0 4050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-rab 3051  df-v 3334  df-dif 3710  df-in 3714  df-ss 3721  df-nul 4051
This theorem is referenced by:  frgrregorufr0  27470
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