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Mirrors > Home > ILE Home > Th. List > nnrei | GIF version |
Description: A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Ref | Expression |
---|---|
nnre.1 | ⊢ 𝐴 ∈ ℕ |
Ref | Expression |
---|---|
nnrei | ⊢ 𝐴 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre.1 | . 2 ⊢ 𝐴 ∈ ℕ | |
2 | nnre 8975 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ ℝ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 ℝcr 7857 ℕcn 8968 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-sep 4143 ax-cnex 7949 ax-resscn 7950 ax-1re 7952 ax-addrcl 7955 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2758 df-in 3155 df-ss 3162 df-int 3867 df-inn 8969 |
This theorem is referenced by: nncni 8978 nnap0i 8999 nnne0i 9000 10re 9452 numlt 9458 numltc 9459 ef01bndlem 11873 pockthi 12470 strleun 12696 strle1g 12698 2strbasg 12711 2stropg 12712 tsetndxnbasendx 12782 plendxnbasendx 12796 dsndxnbasendx 12807 unifndxnbasendx 12817 slotsdifunifndx 12819 |
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