Theorem rabssab 3652

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 2905	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		simpr 476	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜑)
3	2	ss2abi 3637	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝜑}
4	1, 3	eqsstri 3598	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 383   ∈ wcel 1977  {cab 2596  {crab 2900   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-in 3547  df-ss 3554

This theorem is referenced by:  epse  5021  riotasbc  6526  aannenlem2  23888  aalioulem2  23892  ballotlemfmpn  29883  bj-toponss  32241  bj-dmtopon  32242  rencldnfilem  36402  rababg  36898