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Theorem ssmin 4861
 Description: Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
Assertion
Ref Expression
ssmin 𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssmin
StepHypRef Expression
1 ssintab 4859 . 2 (𝐴 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ ∀𝑥((𝐴𝑥𝜑) → 𝐴𝑥))
2 simpl 486 . 2 ((𝐴𝑥𝜑) → 𝐴𝑥)
31, 2mpgbir 1801 1 𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  {cab 2776   ⊆ wss 3883  ∩ cint 4842 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3444  df-in 3890  df-ss 3900  df-int 4843 This theorem is referenced by:  tcid  9183  trclfvlb  14379  trclun  14385  dmtrcl  40498  rntrcl  40499  dfrtrcl5  40500
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