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Theorem nnred 7927
 Description: A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
Hypothesis
Ref Expression
nnred.1 (𝜑𝐴 ∈ ℕ)
Assertion
Ref Expression
nnred (𝜑𝐴 ∈ ℝ)

Proof of Theorem nnred
StepHypRef Expression
1 nnssre 7918 . 2 ℕ ⊆ ℝ
2 nnred.1 . 2 (𝜑𝐴 ∈ ℕ)
31, 2sseldi 2943 1 (𝜑𝐴 ∈ ℝ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  ℝcr 6888  ℕcn 7914 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-cnex 6975  ax-resscn 6976  ax-1re 6978  ax-addrcl 6981 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-int 3616  df-inn 7915 This theorem is referenced by:  qbtwnzlemstep  9103  qbtwnrelemcalc  9110  qbtwnre  9111  flqdiv  9163  modqmulnn  9184  bernneq3  9371  expnbnd  9372  caucvgrelemcau  9579  caucvgre  9580  cvg1nlemcxze  9581  cvg1nlemcau  9583  cvg1nlemres  9584  resqrexlemdecn  9610  resqrexlemga  9621
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