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Mirrors > Home > MPE Home > Th. List > 5re | Structured version Visualization version GIF version |
Description: The number 5 is real. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
5re | ⊢ 5 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 11082 | . 2 ⊢ 5 = (4 + 1) | |
2 | 4re 11097 | . . 3 ⊢ 4 ∈ ℝ | |
3 | 1re 10039 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2, 3 | readdcli 10053 | . 2 ⊢ (4 + 1) ∈ ℝ |
5 | 1, 4 | eqeltri 2697 | 1 ⊢ 5 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 (class class class)co 6650 ℝcr 9935 1c1 9937 + caddc 9939 4c4 11072 5c5 11073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-2 11079 df-3 11080 df-4 11081 df-5 11082 |
This theorem is referenced by: 5cn 11100 6re 11101 6pos 11119 3lt5 11201 2lt5 11202 1lt5 11203 5lt6 11204 4lt6 11205 5lt7 11210 4lt7 11211 5lt8 11217 4lt8 11218 5lt9 11225 4lt9 11226 5lt10OLD 11234 4lt10OLD 11235 5lt10 11677 4lt10 11678 5recm6rec 11686 ef01bndlem 14914 prm23ge5 15520 prmlem1 15814 rmodislmod 18931 sralem 19177 srasca 19181 zlmlem 19865 zlmsca 19869 ppiublem1 24927 ppiub 24929 bposlem3 25011 bposlem4 25012 bposlem5 25013 bposlem6 25014 bposlem8 25016 bposlem9 25017 lgsdir2lem1 25050 gausslemma2dlem4 25094 2lgslem3 25129 cchhllem 25767 ex-id 27291 ex-sqrt 27311 threehalves 29623 resvvsca 29834 zlmds 30008 zlmtset 30009 hgt750lem2 30730 hgt750leme 30736 problem2 31559 problem2OLD 31560 stoweidlem13 40230 31prm 41512 gbegt5 41649 gbowgt5 41650 sbgoldbo 41675 nnsum3primesle9 41682 nnsum4primesodd 41684 evengpop3 41686 |
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