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

Theorem ssintub 4867
 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 4865 . 2 (𝐴 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐴𝑦)
2 sseq2 3969 . . . 4 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
32elrab 3657 . . 3 (𝑦 ∈ {𝑥𝐵𝐴𝑥} ↔ (𝑦𝐵𝐴𝑦))
43simprbi 500 . 2 (𝑦 ∈ {𝑥𝐵𝐴𝑥} → 𝐴𝑦)
51, 4mprgbir 3141 1 𝐴 {𝑥𝐵𝐴𝑥}
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2115  {crab 3130   ⊆ wss 3910  ∩ cint 4849 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rab 3135  df-v 3473  df-in 3917  df-ss 3927  df-int 4850 This theorem is referenced by:  intmin  4869  wuncid  10142  mrcssid  16867  lspssid  19733  lbsextlem3  19908  aspssid  20083  sscls  21640  filufint  22504  spanss2  29107  shsval2i  29149  ococin  29170  chsupsn  29175  sssigagen  31412  dynkin  31434  igenss  35386  pclssidN  37077  dochocss  38548  rgspnssid  39925
 Copyright terms: Public domain W3C validator