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Mirrors > Home > MPE Home > Th. List > 10re | Structured version Visualization version GIF version |
Description: The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
10re | ⊢ ;10 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn 11552 | . 2 ⊢ ;10 ∈ ℕ | |
2 | 1 | nnrei 11067 | 1 ⊢ ;10 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 ℝcr 9973 0cc0 9974 1c1 9975 ;cdc 11531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-dec 11532 |
This theorem is referenced by: 8lt10 11712 7lt10 11713 6lt10 11714 5lt10 11715 4lt10 11716 3lt10 11717 2lt10 11718 1lt10 11719 0.999... 14656 bpoly4 14834 cnfldfun 19806 thlle 20089 bposlem4 25057 bposlem5 25058 dp2cl 29715 dp2lt10 29719 dp2lt 29720 dp2ltsuc 29721 dp2ltc 29722 dpfrac1 29727 dplti 29741 dpgti 29742 dpexpp1 29744 hgt750lem 30857 problem2 31685 bgoldbtbndlem1 42018 tgblthelfgott 42028 tgoldbach 42030 |
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