Theorem ssrabeq 3003

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 3002	. . 3 ⊢ {x ∈ 𝑉 ∣ φ} ⊆ 𝑉
2	1	biantru 286	. 2 ⊢ (𝑉 ⊆ {x ∈ 𝑉 ∣ φ} ↔ (𝑉 ⊆ {x ∈ 𝑉 ∣ φ} ∧ {x ∈ 𝑉 ∣ φ} ⊆ 𝑉))
3		eqss 2937	. 2 ⊢ (𝑉 = {x ∈ 𝑉 ∣ φ} ↔ (𝑉 ⊆ {x ∈ 𝑉 ∣ φ} ∧ {x ∈ 𝑉 ∣ φ} ⊆ 𝑉))
4	2, 3	bitr4i 176	1 ⊢ (𝑉 ⊆ {x ∈ 𝑉 ∣ φ} ↔ 𝑉 = {x ∈ 𝑉 ∣ φ})

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1228  {crab 2288   ⊆ wss 2894

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293  df-in 2901  df-ss 2908

This theorem is referenced by:  difrab0eqim  3265