Theorem ssrab2f 45957

Description: Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
ssrab2f.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
ssrab2f	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴

Proof of Theorem ssrab2f

Step	Hyp	Ref	Expression
1		nfrab1 3434	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
2		ssrab2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	dfss3f 3926	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴)
4		rabidim1 3436	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
5	3, 4	mprgbir 3085	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴

This proof depends on syntax axioms:   ∈ wcel 2145  Ⅎwnfc 2909  {crab 3414   ⊆ wss 3902

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rab 3415  df-ss 3919

This theorem is used by:  dmmptssf  46069  mptssid  46078  fnlimfvre  46510  limsupequzmpt2  46554  liminfequzmpt2  46627  pimltpnff  47539  pimgtmnff  47558  smflimlem2  47608  smflim  47613  smfsupxr  47652  smfpimne2  47676