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Theorem elnpi 8866
 Description: Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elnpi
Distinct variable group:   ,,

Proof of Theorem elnpi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2965 . 2
2 simpl1 961 . 2
3 psseq2 3436 . . . . . 6
4 psseq1 3435 . . . . . 6
53, 4anbi12d 693 . . . . 5
6 eleq2 2498 . . . . . . . . 9
76imbi2d 309 . . . . . . . 8
87albidv 1636 . . . . . . 7
9 rexeq 2906 . . . . . . 7
108, 9anbi12d 693 . . . . . 6
1110raleqbi1dv 2913 . . . . 5
125, 11anbi12d 693 . . . 4
13 df-np 8859 . . . 4
1412, 13elab2g 3085 . . 3
15 id 21 . . . . . 6
16153expib 1157 . . . . 5
17 3simpc 957 . . . . 5
1816, 17impbid1 196 . . . 4
1918anbi1d 687 . . 3
2014, 19bitrd 246 . 2
211, 2, 20pm5.21nii 344 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937  wal 1550   wceq 1653   wcel 1726  wral 2706  wrex 2707  cvv 2957   wpss 3322  c0 3629   class class class wbr 4213  cnq 8728   cltq 8734  cnp 8735 This theorem is referenced by:  prn0  8867  prpssnq  8868  prcdnq  8871  prnmax  8873 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-v 2959  df-in 3328  df-ss 3335  df-pss 3337  df-np 8859
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