Theorem rabssab 3184

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 2425	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		simpr 109	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜑)
3	2	ss2abi 3169	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝜑}
4	1, 3	eqsstri 3129	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 103   ∈ wcel 1480  {cab 2125  {crab 2420   ⊆ wss 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-in 3077  df-ss 3084

This theorem is referenced by:  epse  4264  riotasbc  5745  genipv  7317  toponsspwpwg  12189  dmtopon  12190