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Theorem pnfnre 9127
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 6545 . . . 4  |-  -.  ~P U. CC  e.  CC
2 df-pnf 9122 . . . . 5  |-  +oo  =  ~P U. CC
32eleq1i 2499 . . . 4  |-  (  +oo  e.  CC  <->  ~P U. CC  e.  CC )
41, 3mtbir 291 . . 3  |-  -.  +oo  e.  CC
5 recn 9080 . . 3  |-  (  +oo  e.  RR  ->  +oo  e.  CC )
64, 5mto 169 . 2  |-  -.  +oo  e.  RR
76nelir 2698 1  |-  +oo  e/  RR
Colors of variables: wff set class
Syntax hints:    e. wcel 1725    e/ wnel 2600   ~Pcpw 3799   U.cuni 4015   CCcc 8988   RRcr 8989    +oocpnf 9117
This theorem is referenced by:  renepnf  9132  ltxrlt  9146  xrltnr  10720  pnfnlt  10725  hashclb  11641  hasheq0  11644  pcgcd1  13250  pc2dvds  13252  ramtcl2  13379  odhash3  15210  xrsdsreclblem  16744  pnfnei  17284  iccpnfcnv  18969  i1f0rn  19574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821  df-uni 4016  df-pnf 9122
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