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Theorem prnmax 9371
 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 2494 . . . . 5
21anbi2d 708 . . . 4
3 breq1 4369 . . . . 5
43rexbidv 2878 . . . 4
52, 4imbi12d 321 . . 3
6 elnpi 9364 . . . . . 6
76simprbi 465 . . . . 5
87r19.21bi 2734 . . . 4
98simprd 464 . . 3
105, 9vtoclg 3082 . 2
1110anabsi7 826 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982  wal 1435   wceq 1437   wcel 1872  wral 2714  wrex 2715  cvv 3022   wpss 3380  c0 3704   class class class wbr 4366  cnq 9228   cltq 9234  cnp 9235 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-np 9357 This theorem is referenced by:  npomex  9372  prnmadd  9373  genpnmax  9383  1idpr  9405  ltexprlem4  9415  reclem3pr  9425  suplem1pr  9428
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