Theorem ssab2 4041

Description: Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)

Assertion

Ref	Expression
ssab2	⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssab2

Step	Hyp	Ref	Expression
1		simpl 487	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴)
2	1	abssi 4030	1 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴

Syntax hints:   ∧ wa 400   ∈ wcel 2149  {cab 2747   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ss 3930

This theorem is referenced by:  zfausab  5303  exss  5445  dmopabss  5909  rnopabss  5946  isf32lem9  10344  psubspset  40407  psubclsetN  40599