Theorem rabssab 3353

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 2624	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
2		simpr 447	. . 3 ⊢ ((x ∈ A ∧ φ) → φ)
3	2	ss2abi 3339	. 2 ⊢ {x ∣ (x ∈ A ∧ φ)} ⊆ {x ∣ φ}
4	1, 3	eqsstri 3302	1 ⊢ {x ∈ A ∣ φ} ⊆ {x ∣ φ}

This proof depends on syntax axioms:   ∧ wa 358   ∈ wcel 1710  {cab 2339  {crab 2619   ⊆ wss 3258

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is used by: (None)