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Theorem ssmin 4928
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 4926 . 2 (𝐴 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ ∀𝑥((𝐴𝑥𝜑) → 𝐴𝑥))
2 simpl 483 . 2 ((𝐴𝑥𝜑) → 𝐴𝑥)
31, 2mpgbir 1801 1 𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  {cab 2713  wss 3910   cint 4907
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-v 3447  df-in 3917  df-ss 3927  df-int 4908
This theorem is referenced by:  tcid  9674  trclfvlb  14892  trclun  14898  dmtrcl  41880  rntrcl  41881  dfrtrcl5  41882
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