Theorem difrab0eq 4420

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 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅ ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑})

Distinct variable group:   𝑥,𝑉

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem difrab0eq

Step	Hyp	Ref	Expression
1		ssdif0 4316	. 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅)
2		ssrabeq 4034	. 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑})
3	1, 2	bitr3i 277	1 ⊢ ((𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅ ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑})

Syntax hints:   ↔ wb 206   = wceq 1541  {crab 3397   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-ss 3916  df-nul 4284

This theorem is referenced by:  frgrregorufr0  30348