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Theorem xnn0xr 9239
Description: An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
xnn0xr (𝐴 ∈ ℕ0*𝐴 ∈ ℝ*)

Proof of Theorem xnn0xr
StepHypRef Expression
1 elxnn0 9236 . 2 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
2 nn0re 9180 . . . 4 (𝐴 ∈ ℕ0𝐴 ∈ ℝ)
32rexrd 8002 . . 3 (𝐴 ∈ ℕ0𝐴 ∈ ℝ*)
4 pnfxr 8005 . . . 4 +∞ ∈ ℝ*
5 eleq1 2240 . . . 4 (𝐴 = +∞ → (𝐴 ∈ ℝ* ↔ +∞ ∈ ℝ*))
64, 5mpbiri 168 . . 3 (𝐴 = +∞ → 𝐴 ∈ ℝ*)
73, 6jaoi 716 . 2 ((𝐴 ∈ ℕ0𝐴 = +∞) → 𝐴 ∈ ℝ*)
81, 7sylbi 121 1 (𝐴 ∈ ℕ0*𝐴 ∈ ℝ*)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 708   = wceq 1353  wcel 2148  +∞cpnf 7984  *cxr 7986  0cn0 9171  0*cxnn0 9234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-un 4432  ax-cnex 7898  ax-resscn 7899  ax-1re 7901  ax-addrcl 7904  ax-rnegex 7916
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-pnf 7989  df-xr 7991  df-inn 8915  df-n0 9172  df-xnn0 9235
This theorem is referenced by:  xnn0xrnemnf  9246  xnn0dcle  9797  xnn0letri  9798
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