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Theorem pnfnre 8874
Description: Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
pnfnre  |-  +oo  e/  RR

Proof of Theorem pnfnre
StepHypRef Expression
1 pwuninel 6300 . . . 4  |-  -.  ~P U. CC  e.  CC
2 df-pnf 8869 . . . . 5  |-  +oo  =  ~P U. CC
32eleq1i 2346 . . . 4  |-  (  +oo  e.  CC  <->  ~P U. CC  e.  CC )
41, 3mtbir 290 . . 3  |-  -.  +oo  e.  CC
5 recn 8827 . . 3  |-  (  +oo  e.  RR  ->  +oo  e.  CC )
64, 5mto 167 . 2  |-  -.  +oo  e.  RR
7 df-nel 2449 . 2  |-  (  +oo  e/  RR  <->  -.  +oo  e.  RR )
86, 7mpbir 200 1  |-  +oo  e/  RR
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1684    e/ wnel 2447   ~Pcpw 3625   U.cuni 3827   CCcc 8735   RRcr 8736    +oocpnf 8864
This theorem is referenced by:  renepnf  8879  ltxrlt  8893  xrltnr  10462  pnfnlt  10467  hashclb  11352  hasheq0  11353  pcgcd1  12929  pc2dvds  12931  ramtcl2  13058  odhash3  14887  xrsdsreclblem  16417  pnfnei  16950  iccpnfcnv  18442  i1f0rn  19037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-pnf 8869
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