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Theorem xnn0xr 9064
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 9061 . 2 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
2 nn0re 9005 . . . 4 (𝐴 ∈ ℕ0𝐴 ∈ ℝ)
32rexrd 7834 . . 3 (𝐴 ∈ ℕ0𝐴 ∈ ℝ*)
4 pnfxr 7837 . . . 4 +∞ ∈ ℝ*
5 eleq1 2202 . . . 4 (𝐴 = +∞ → (𝐴 ∈ ℝ* ↔ +∞ ∈ ℝ*))
64, 5mpbiri 167 . . 3 (𝐴 = +∞ → 𝐴 ∈ ℝ*)
73, 6jaoi 705 . 2 ((𝐴 ∈ ℕ0𝐴 = +∞) → 𝐴 ∈ ℝ*)
81, 7sylbi 120 1 (𝐴 ∈ ℕ0*𝐴 ∈ ℝ*)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 697   = wceq 1331  wcel 1480  +∞cpnf 7816  *cxr 7818  0cn0 8996  0*cxnn0 9059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-un 4358  ax-cnex 7730  ax-resscn 7731  ax-1re 7733  ax-addrcl 7736  ax-rnegex 7748
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3740  df-int 3775  df-pnf 7821  df-xr 7823  df-inn 8740  df-n0 8997  df-xnn0 9060
This theorem is referenced by:  xnn0xrnemnf  9071
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