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Mirrors > Home > MPE Home > Th. List > nnrei | Structured version Visualization version 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 10904 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ℝcr 9814 ℕcn 10897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 |
This theorem is referenced by: nncni 10907 nnne0i 10932 10re 11393 numlt 11403 numltc 11404 faclbnd4lem1 12942 ef01bndlem 14753 dvdslelem 14869 divalglem6 14959 pockthi 15449 modsubi 15614 prmlem1 15652 prmlem2 15665 strlemor1 15796 strleun 15799 strle1 15800 oppchomfval 16197 oppcbas 16201 rescco 16315 opprlem 18451 sralem 18998 opsrbaslem 19298 opsrbaslemOLD 19299 zlmlem 19684 znbaslem 19705 znbaslemOLD 19706 tnglem 22254 log2ublem1 24473 log2ublem2 24474 log2ub 24476 bpos1lem 24807 bposlem8 24816 bposlem9 24817 ttgval 25555 ttglem 25556 cchhllem 25567 slotsbaseefdif 25672 structvtxvallem 25697 structvtxval 25698 structiedg0val 25699 lmat22e12 29213 lmat22e21 29214 lmat22e22 29215 ballotlem2 29877 ballotlem5 29888 ballotth 29926 cnndvlem1 31698 hlhilslem 36248 jm2.27dlem2 36595 bgoldbtbndlem1 40221 tgblthelfgott 40229 tgoldbachlt 40230 tgblthelfgottOLD 40236 tgoldbachltOLD 40237 |
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