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Theorem ssmin 3827
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 3825 . 2 (𝐴 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ ∀𝑥((𝐴𝑥𝜑) → 𝐴𝑥))
2 simpl 108 . 2 ((𝐴𝑥𝜑) → 𝐴𝑥)
31, 2mpgbir 1433 1 𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  {cab 2143  wss 3102   cint 3808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-int 3809
This theorem is referenced by: (None)
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