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Theorem elnpi 9387
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 3118 . 2
2 simpl1 999 . 2
3 psseq2 3591 . . . . . 6
4 psseq1 3590 . . . . . 6
53, 4anbi12d 710 . . . . 5
6 eleq2 2530 . . . . . . . . 9
76imbi2d 316 . . . . . . . 8
87albidv 1713 . . . . . . 7
9 rexeq 3055 . . . . . . 7
108, 9anbi12d 710 . . . . . 6
1110raleqbi1dv 3062 . . . . 5
125, 11anbi12d 710 . . . 4
13 df-np 9380 . . . 4
1412, 13elab2g 3248 . . 3
15 id 22 . . . . . 6
16153expib 1199 . . . . 5
17 3simpc 995 . . . . 5
1816, 17impbid1 203 . . . 4
1918anbi1d 704 . . 3
2014, 19bitrd 253 . 2
211, 2, 20pm5.21nii 353 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  A.wal 1393  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808   cvv 3109  C.wpss 3476   c0 3784   class class class wbr 4452   cnq 9251   cltq 9257   cnp 9258
This theorem is referenced by:  prn0  9388  prpssnq  9389  prcdnq  9392  prnmax  9394
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-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-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-pss 3491  df-np 9380
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