Theorem ssab2 3985

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 486	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴)
2	1	abssi 3976	1 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴

Syntax hints:   ∧ wa 399   ∈ wcel 2111  {cab 2735   ⊆ wss 3860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3867  df-ss 3877

This theorem is referenced by:  ssrab2OLD  3987  zfausab  5204  exss  5327  dmopabss  5764  fabexg  7650  isf32lem9  9834  psubspset  37355  psubclsetN  37547