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Theorem pnfnre 8890
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 6316 . . . 4  |-  -.  ~P U. CC  e.  CC
2 df-pnf 8885 . . . . 5  |-  +oo  =  ~P U. CC
32eleq1i 2359 . . . 4  |-  (  +oo  e.  CC  <->  ~P U. CC  e.  CC )
41, 3mtbir 290 . . 3  |-  -.  +oo  e.  CC
5 recn 8843 . . 3  |-  (  +oo  e.  RR  ->  +oo  e.  CC )
64, 5mto 167 . 2  |-  -.  +oo  e.  RR
7 df-nel 2462 . 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 1696    e/ wnel 2460   ~Pcpw 3638   U.cuni 3843   CCcc 8751   RRcr 8752    +oocpnf 8880
This theorem is referenced by:  renepnf  8895  ltxrlt  8909  xrltnr  10478  pnfnlt  10483  hashclb  11368  hasheq0  11369  pcgcd1  12945  pc2dvds  12947  ramtcl2  13074  odhash3  14903  xrsdsreclblem  16433  pnfnei  16966  iccpnfcnv  18458  i1f0rn  19053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660  df-uni 3844  df-pnf 8885
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