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Mirrors > Home > MPE Home > Th. List > 4pos | Structured version Visualization version GIF version |
Description: The number 4 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
4pos | ⊢ 0 < 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3re 11711 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1re 10635 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 3pos 11736 | . . 3 ⊢ 0 < 3 | |
4 | 0lt1 11156 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 11176 | . 2 ⊢ 0 < (3 + 1) |
6 | df-4 11696 | . 2 ⊢ 4 = (3 + 1) | |
7 | 5, 6 | breqtrri 5085 | 1 ⊢ 0 < 4 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5058 (class class class)co 7150 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 3c3 11687 4c4 11688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-2 11694 df-3 11695 df-4 11696 |
This theorem is referenced by: 4ne0 11739 5pos 11740 div4p1lem1div2 11886 fldiv4p1lem1div2 13199 iexpcyc 13563 discr 13595 faclbnd2 13645 sqrt2gt1lt2 14628 flodddiv4 15758 pcoass 23622 csbren 23996 minveclem2 24023 dveflem 24570 sincos4thpi 25093 log2cnv 25516 chtublem 25781 bposlem6 25859 gausslemma2dlem0d 25929 2sqlem11 25999 chebbnd1lem3 26041 chebbnd1 26042 pntibndlem1 26159 pntlemb 26167 pntlemg 26168 pntlemr 26172 pntlemf 26175 usgrexmplef 27035 upgr4cycl4dv4e 27958 minvecolem2 28646 minvecolem3 28647 normlem6 28886 sqsscirc1 31146 hgt750lem 31917 limclner 41925 stoweid 42342 stirlinglem10 42362 stirlinglem12 42364 bgoldbtbndlem3 43966 itsclc0yqsollem2 44744 itscnhlinecirc02plem1 44763 |
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