Theorem rabssab 3021

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 2309	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
2		simpr 103	. . 3 ⊢ ((x ∈ A ∧ φ) → φ)
3	2	ss2abi 3006	. 2 ⊢ {x ∣ (x ∈ A ∧ φ)} ⊆ {x ∣ φ}
4	1, 3	eqsstri 2969	1 ⊢ {x ∈ A ∣ φ} ⊆ {x ∣ φ}

Syntax hints:   ∧ wa 97   ∈ wcel 1390  {cab 2023  {crab 2304   ⊆ wss 2911

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-in 2918  df-ss 2925

This theorem is referenced by:  epse  4064  riotasbc  5426  genipv  6492