Theorem ssmin 3946

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 3944	. 2 ⊢ (A ⊆ ∩{x ∣ (A ⊆ x ∧ φ)} ↔ ∀x((A ⊆ x ∧ φ) → A ⊆ x))
2		simpl 443	. 2 ⊢ ((A ⊆ x ∧ φ) → A ⊆ x)
3	1, 2	mpgbir 1550	1 ⊢ A ⊆ ∩{x ∣ (A ⊆ x ∧ φ)}

Syntax hints:   → wi 4   ∧ wa 358  {cab 2339   ⊆ wss 3258  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is referenced by:  clos1base  5879