Theorem ssrabeq 3080

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 3079	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉
2	1	biantru 296	. 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ∧ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉))
3		eqss 3014	. 2 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ∧ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉))
4	2, 3	bitr4i 185	1 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑})

Syntax hints:   ∧ wa 102   ↔ wb 103   = wceq 1284  {crab 2352   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-in 2979  df-ss 2986

This theorem is referenced by:  difrab0eqim  3310