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Theorem ssintub 4730
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 4728 . 2 (𝐴 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐴𝑦)
2 sseq2 3846 . . . 4 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
32elrab 3572 . . 3 (𝑦 ∈ {𝑥𝐵𝐴𝑥} ↔ (𝑦𝐵𝐴𝑦))
43simprbi 492 . 2 (𝑦 ∈ {𝑥𝐵𝐴𝑥} → 𝐴𝑦)
51, 4mprgbir 3109 1 𝐴 {𝑥𝐵𝐴𝑥}
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  {crab 3094  wss 3792   cint 4712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-in 3799  df-ss 3806  df-int 4713
This theorem is referenced by:  intmin  4732  wuncid  9902  mrcssid  16674  lspssid  19391  lbsextlem3  19568  aspssid  19741  sscls  21279  filufint  22143  spanss2  28793  shsval2i  28835  ococin  28856  chsupsn  28861  sssigagen  30814  dynkin  30836  igenss  34494  pclssidN  36058  dochocss  37529  rgspnssid  38713
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