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Theorem ssintub 4970
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 4968 . 2 (𝐴 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐴𝑦)
2 sseq2 4021 . . . 4 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
32elrab 3694 . . 3 (𝑦 ∈ {𝑥𝐵𝐴𝑥} ↔ (𝑦𝐵𝐴𝑦))
43simprbi 496 . 2 (𝑦 ∈ {𝑥𝐵𝐴𝑥} → 𝐴𝑦)
51, 4mprgbir 3065 1 𝐴 {𝑥𝐵𝐴𝑥}
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  {crab 3432  wss 3962   cint 4950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rab 3433  df-v 3479  df-ss 3979  df-int 4951
This theorem is referenced by:  intmin  4972  cofon2  8709  naddunif  8729  wuncid  10780  mrcssid  17661  rgspnssid  20630  lspssid  21000  lbsextlem3  21179  aspssid  21915  sscls  23079  filufint  23943  spanss2  31373  shsval2i  31415  ococin  31436  chsupsn  31441  fldgenssid  33294  sssigagen  34125  dynkin  34147  igenss  38048  pclssidN  39877  dochocss  41348  intubeu  48772
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