Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  supub Structured version   Unicode version

Theorem supub 7456
 Description: A supremum is an upper bound. See also supcl 7455 and suplub 7457. This proof demonstrates how to expand an iota-based definition (df-iota 5410) using riotacl2 6555. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Hypotheses
Ref Expression
supmo.1
supcl.2
Assertion
Ref Expression
supub
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem supub
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . . . . 6
21a1i 11 . . . . 5
32ss2rabi 3417 . . . 4
4 supmo.1 . . . . . 6
5 supcl.2 . . . . . 6
64, 5supval2 7452 . . . . 5
74, 5supeu 7451 . . . . . 6
8 riotacl2 6555 . . . . . 6
97, 8syl 16 . . . . 5
106, 9eqeltrd 2509 . . . 4
113, 10sseldi 3338 . . 3
12 breq2 4208 . . . . . . . 8
1312notbid 286 . . . . . . 7
1413cbvralv 2924 . . . . . 6
15 breq1 4207 . . . . . . . 8
1615notbid 286 . . . . . . 7
1716ralbidv 2717 . . . . . 6
1814, 17syl5bb 249 . . . . 5
1918elrab 3084 . . . 4
2019simprbi 451 . . 3
2111, 20syl 16 . 2
22 breq2 4208 . . . 4
2322notbid 286 . . 3
2423rspccv 3041 . 2
2521, 24syl 16 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  wreu 2699  crab 2701   class class class wbr 4204   wor 4494  crio 6534  csup 7437 This theorem is referenced by:  suplub2  7458  supmax  7462  supiso  7469  suprub  9961  infmrlb  9981  suprzub  10559  supxrun  10886  supxrub  10895  infmxrlb  10904  dgrub  20145  supssd  24090  ssnnssfz  24140  ballotlemimin  24755  ballotlemfrcn0  24779  wsuclb  25571  itg2addnclem  26246  supubt  26432 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-po 4495  df-so 4496  df-iota 5410  df-riota 6541  df-sup 7438
 Copyright terms: Public domain W3C validator