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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 4010 . . . 4 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
32elrab 3692 . . 3 (𝑦 ∈ {𝑥𝐵𝐴𝑥} ↔ (𝑦𝐵𝐴𝑦))
43simprbi 496 . 2 (𝑦 ∈ {𝑥𝐵𝐴𝑥} → 𝐴𝑦)
51, 4mprgbir 3068 1 𝐴 {𝑥𝐵𝐴𝑥}
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  {crab 3436  wss 3951   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-ss 3968  df-int 4947
This theorem is referenced by:  intmin  4968  cofon2  8711  naddunif  8731  wuncid  10783  mrcssid  17660  rgspnssid  20614  lspssid  20983  lbsextlem3  21162  aspssid  21898  sscls  23064  filufint  23928  spanss2  31364  shsval2i  31406  ococin  31427  chsupsn  31432  fldgenssid  33315  sssigagen  34146  dynkin  34168  igenss  38069  pclssidN  39897  dochocss  41368  intubeu  48873
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