Theorem ssab2 4007

Description: Subclass relation for the restriction of a class abstraction. Prefer using the more natural statement ssrab2 4008. (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 488	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴)
2	1	abssi 3996	1 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2147  {cab 2741   ⊆ wss 3900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901

This theorem is used by:  sepab  5301  exss  5438  dmopabss  5912  rnopabss  5949  isf32lem9  10478  psubspset  41067  psubclsetN  41259