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Theorem ssmin 4468
 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 4466 . 2 (𝐴 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ ∀𝑥((𝐴𝑥𝜑) → 𝐴𝑥))
2 simpl 473 . 2 ((𝐴𝑥𝜑) → 𝐴𝑥)
31, 2mpgbir 1723 1 𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  {cab 2607   ⊆ wss 3560  ∩ cint 4447 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-v 3192  df-in 3567  df-ss 3574  df-int 4448 This theorem is referenced by:  tcid  8575  trclfvlb  13699  trclun  13705  dmtrcl  37454  rntrcl  37455  dfrtrcl5  37456
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