Theorem ssrabeq 3257

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 3255	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉
2	1	biantru 302	. 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ∧ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉))
3		eqss 3185	. 2 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ∧ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉))
4	2, 3	bitr4i 187	1 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑})

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  {crab 2472   ⊆ wss 3144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-in 3150  df-ss 3157

This theorem is referenced by:  difrab0eqim  3504