Theorem ssmin 4878

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 4876	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} ↔ ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥))
2		simpl 486	. 2 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥)
3	1, 2	mpgbir 1807	1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)}

Syntax hints:   → wi 4   ∧ wa 399  {cab 2714   ⊆ wss 3866  ∩ cint 4859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-v 3410  df-in 3873  df-ss 3883  df-int 4860

This theorem is referenced by:  tcid  9355  trclfvlb  14571  trclun  14577  dmtrcl  40911  rntrcl  40912  dfrtrcl5  40913