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

Theorem ssintub 4966
Description: Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
Assertion
Ref Expression
ssintub 𝐴 {𝑥𝐵𝐴𝑥}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssintub
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssint 4964 . 2 (𝐴 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐴𝑦)
2 sseq2 4005 . . . 4 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
32elrab 3680 . . 3 (𝑦 ∈ {𝑥𝐵𝐴𝑥} ↔ (𝑦𝐵𝐴𝑦))
43simprbi 495 . 2 (𝑦 ∈ {𝑥𝐵𝐴𝑥} → 𝐴𝑦)
51, 4mprgbir 3058 1 𝐴 {𝑥𝐵𝐴𝑥}
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  {crab 3419  wss 3946   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rab 3420  df-v 3464  df-ss 3963  df-int 4947
This theorem is referenced by:  intmin  4968  cofon2  8695  naddunif  8715  wuncid  10777  mrcssid  17625  lspssid  20958  lbsextlem3  21137  aspssid  21871  sscls  23048  filufint  23912  spanss2  31275  shsval2i  31317  ococin  31338  chsupsn  31343  fldgenssid  33168  sssigagen  33991  dynkin  34013  igenss  37776  pclssidN  39607  dochocss  41078  rgspnssid  42868  intubeu  48346
  Copyright terms: Public domain W3C validator