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Theorem supmaxOLD 7946
 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.) Obsolete version of supmax 7944 as of 30-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
supmaxOLD.1
supmaxOLD.2
supmaxOLD.3
supmaxOLD.4
Assertion
Ref Expression
supmaxOLD
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem supmaxOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supmaxOLD.3 . . 3
2 supmaxOLD.1 . . . 4
3 supmaxOLD.2 . . . . 5
4 supmaxOLD.4 . . . . . 6
54ralrimiva 2871 . . . . 5
6 supmaxlemOLD 7945 . . . . 5
73, 1, 5, 6syl3anc 1228 . . . 4
82, 7supub 7939 . . 3
91, 8mpd 15 . 2
102, 7supnub 7942 . . 3
113, 5, 10mp2and 679 . 2
122, 7supcl 7938 . . 3
13 sotrieq2 4833 . . 3
142, 12, 3, 13syl12anc 1226 . 2
159, 11, 14mpbir2and 922 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808   class class class wbr 4452  Orwor 4804  supcsup 7920 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-po 4805  df-so 4806  df-iota 5556  df-riota 6257  df-sup 7921
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