Theorem ssmin 4909

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 4907	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} ↔ ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥))
2		simpl 482	. 2 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥)
3	1, 2	mpgbir 1801	1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)}

Syntax hints:   → wi 4   ∧ wa 395  {cab 2714   ⊆ wss 3889  ∩ cint 4889

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3431  df-ss 3906  df-int 4890

This theorem is referenced by:  tcid  9658  trclfvlb  14970  trclun  14976  tz9.1regs  35278  tz9.1tco  36665  dfttc3gw  36705  dmtrcl  44054  rntrcl  44055  dfrtrcl5  44056