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Theorem prnmax 9301
 Description: A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prnmax
Distinct variable groups:   ,   ,

Proof of Theorem prnmax
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eleq1 2526 . . . . 5
21anbi2d 703 . . . 4
3 breq1 4412 . . . . 5
43rexbidv 2875 . . . 4
52, 4imbi12d 320 . . 3
6 elnpi 9294 . . . . . 6
76simprbi 464 . . . . 5
87r19.21bi 2922 . . . 4
98simprd 463 . . 3
105, 9vtoclg 3139 . 2
1110anabsi7 815 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  /\w3a 965  A.wal 1368  =wceq 1370  e.wcel 1758  A.wral 2800  E.wrex 2801   cvv 3081  C.wpss 3443   c0 3751   class class class wbr 4409   cnq 9156   cltq 9162   cnp 9163 This theorem is referenced by:  npomex  9302  prnmadd  9303  genpnmax  9313  1idpr  9335  ltexprlem4  9345  reclem3pr  9355  suplem1pr  9358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3083  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3752  df-if 3906  df-sn 3994  df-pr 3996  df-op 4000  df-br 4410  df-np 9287
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