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Theorem supmaxlem 7469
 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.)
Assertion
Ref Expression
supmaxlem
Distinct variable groups:   ,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,)

Proof of Theorem supmaxlem
StepHypRef Expression
1 breq2 4216 . . . . . . 7
21rspcev 3052 . . . . . 6
32ex 424 . . . . 5
43ralrimivw 2790 . . . 4
5 breq2 4216 . . . . . . 7
65notbid 286 . . . . . 6
76cbvralv 2932 . . . . 5
87biimpi 187 . . . 4
94, 8anim12ci 551 . . 3
10 breq1 4215 . . . . . . 7
1110notbid 286 . . . . . 6
1211ralbidv 2725 . . . . 5
13 breq2 4216 . . . . . . 7
1413imbi1d 309 . . . . . 6
1514ralbidv 2725 . . . . 5
1612, 15anbi12d 692 . . . 4
1716rspcev 3052 . . 3
189, 17sylan2 461 . 2
19183impb 1149 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  wrex 2706   class class class wbr 4212 This theorem is referenced by:  supmax  7470 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-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  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-br 4213
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