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Theorem supmaxlemOLD 7945
Description: A set that contains the greatest element satisfies the antecedent in supremum theorems. This allows to be used in some situations without the completeness axiom. (Contributed by Jeff Hoffman, 17-Jun-2008.) Obsolete as of 30-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
supmaxlemOLD
Distinct variable groups:   ,   , , ,   , , ,   , , ,

Proof of Theorem supmaxlemOLD
StepHypRef Expression
1 breq2 4456 . . . . . . 7
21rspcev 3210 . . . . . 6
32ex 434 . . . . 5
43ralrimivw 2872 . . . 4
5 breq2 4456 . . . . . . 7
65notbid 294 . . . . . 6
76cbvralv 3084 . . . . 5
87biimpi 194 . . . 4
94, 8anim12ci 567 . . 3
10 breq1 4455 . . . . . . 7
1110notbid 294 . . . . . 6
1211ralbidv 2896 . . . . 5
13 breq2 4456 . . . . . . 7
1413imbi1d 317 . . . . . 6
1514ralbidv 2896 . . . . 5
1612, 15anbi12d 710 . . . 4
1716rspcev 3210 . . 3
189, 17sylan2 474 . 2
19183impb 1192 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808   class class class wbr 4452
This theorem is referenced by:  supmaxOLD  7946
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-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  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-br 4453
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