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

Theorem supex 8410
Description: A supremum is a set. (Contributed by NM, 22-May-1999.)
Hypothesis
Ref Expression
supex.1 𝑅 Or 𝐴
Assertion
Ref Expression
supex sup(𝐵, 𝐴, 𝑅) ∈ V

Proof of Theorem supex
StepHypRef Expression
1 supex.1 . 2 𝑅 Or 𝐴
2 id 22 . . 3 (𝑅 Or 𝐴𝑅 Or 𝐴)
32supexd 8400 . 2 (𝑅 Or 𝐴 → sup(𝐵, 𝐴, 𝑅) ∈ V)
41, 3ax-mp 5 1 sup(𝐵, 𝐴, 𝑅) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  Vcvv 3231   Or wor 5063  supcsup 8387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rmo 2949  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-po 5064  df-so 5065  df-sup 8389
This theorem is referenced by:  limsupgval  14251  limsupgre  14256  gcdval  15265  pczpre  15599  prmreclem1  15667  prdsdsfn  16172  prdsdsval  16185  xrge0tsms2  22685  mbfsup  23476  mbfinf  23477  itg2val  23540  itg2monolem1  23562  itg2mono  23565  mdegval  23868  mdegxrf  23873  plyeq0lem  24011  dgrval  24029  nmooval  27746  nmopval  28843  nmfnval  28863  lmdvg  30127  esumval  30236  erdszelem3  31301  erdszelem6  31304  gtinfOLD  32439  supcnvlimsup  40290  limsuplt2  40303  liminfval  40309  limsupge  40311  liminflelimsuplem  40325  fourierdlem79  40720  sge0val  40901  sge0tsms  40915  smflimsuplem1  41347  smflimsuplem2  41348  smflimsuplem4  41350
  Copyright terms: Public domain W3C validator