Theorem rabssab 4108

Description: A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
rabssab	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}

Proof of Theorem rabssab

Step	Hyp	Ref	Expression
1		df-rab 3444	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		simpr 484	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜑)
3	2	ss2abi 4090	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝜑}
4	1, 3	eqsstri 4043	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 395   ∈ wcel 2108  {cab 2717  {crab 3443   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-rab 3444  df-ss 3993

This theorem is referenced by:  epse  5682  riotasbc  7423  cshwsexa  14872  toponsspwpw  22949  dmtopon  22950  aannenlem2  26389  aalioulem2  26393  ballotlemfmpn  34459  rencldnfilem  42776  rmxyelqirr  42866  rababg  43536