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Theorem supmax 7470
 Description: The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.)
Hypotheses
Ref Expression
supmax.1
supmax.2
supmax.3
supmax.4
Assertion
Ref Expression
supmax
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem supmax
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supmax.3 . . 3
2 supmax.1 . . . 4
3 supmax.2 . . . . 5
4 supmax.4 . . . . . 6
54ralrimiva 2789 . . . . 5
6 supmaxlem 7469 . . . . 5
73, 1, 5, 6syl3anc 1184 . . . 4
82, 7supub 7464 . . 3
91, 8mpd 15 . 2
102, 7supnub 7467 . . 3
113, 5, 10mp2and 661 . 2
122, 7supcl 7463 . . 3
13 sotrieq2 4531 . . 3
142, 12, 3, 13syl12anc 1182 . 2
159, 11, 14mpbir2and 889 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2705  wrex 2706   class class class wbr 4212   wor 4502  csup 7445 This theorem is referenced by:  suppr  7473  lbinfm  9961  ramcl2lem  13377  gsumesum  24451  ballotlemirc  24789  supfz  25199  inffz  25200  mblfinlem2  26244 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 2417 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-po 4503  df-so 4504  df-iota 5418  df-riota 6549  df-sup 7446
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