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Theorem supeu 7173
 Description: A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.)
Hypotheses
Ref Expression
supmo.1
supeu.2
Assertion
Ref Expression
supeu
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem supeu
StepHypRef Expression
1 supeu.2 . 2
2 supmo.1 . . 3
32supmo 7171 . 2
4 reu5 2728 . 2
51, 3, 4sylanbrc 648 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wa 360  wral 2518  wrex 2519  wreu 2520  wrmo 2521   class class class wbr 3997   wor 4285 This theorem is referenced by:  supval2  7174  eqsup  7175  supcl  7177  supub  7178  suplub  7179  fisup2g  7185  fisupcl  7186  lbinfm  9675  supeutOLD  25790 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-po 4286  df-so 4287
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