Theorem ssrabeq 3651

Description: If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)

Assertion

Ref	Expression
ssrabeq	⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑})

Distinct variable group:   𝑥,𝑉

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssrabeq

Step	Hyp	Ref	Expression
1		ssrab2 3650	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉
2	1	biantru 525	. 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ∧ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉))
3		eqss 3583	. 2 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ∧ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉))
4	2, 3	bitr4i 266	1 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑})

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  {crab 2900   ⊆ wss 3540

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-in 3547  df-ss 3554

This theorem is referenced by:  difrab0eq  3990