MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssmin Structured version   Visualization version   GIF version

Theorem ssmin 4972
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 4970 . 2 (𝐴 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ ∀𝑥((𝐴𝑥𝜑) → 𝐴𝑥))
2 simpl 482 . 2 ((𝐴𝑥𝜑) → 𝐴𝑥)
31, 2mpgbir 1796 1 𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  {cab 2712  wss 3963   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-int 4952
This theorem is referenced by:  tcid  9777  trclfvlb  15044  trclun  15050  dmtrcl  43617  rntrcl  43618  dfrtrcl5  43619
  Copyright terms: Public domain W3C validator