Theorem ssmin 4934

Description: Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)

Assertion

Ref	Expression
ssmin	⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)}

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssmin

Step	Hyp	Ref	Expression
1		ssintab 4932	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} ↔ ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥))
2		simpl 488	. 2 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥)
3	1, 2	mpgbir 1832	1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)}

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  {cab 2743   ⊆ wss 3906  ∩ cint 4914

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

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 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-v 3459  df-ss 3923  df-int 4915

This theorem is used by:  tcid  9713  trclfvlb  15069  trclun  15075  tz9.1regs  35604  tz9.1tco  37051  dfttc3gw  37091  dmtrcl  44411  rntrcl  44412  dfrtrcl5  44413