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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 12314 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1re 11236 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 3pos 12339 | . . 3 ⊢ 0 < 3 | |
4 | 0lt1 11758 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 11778 | . 2 ⊢ 0 < (3 + 1) |
6 | df-4 12299 | . 2 ⊢ 4 = (3 + 1) | |
7 | 5, 6 | breqtrri 5169 | 1 ⊢ 0 < 4 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5142 (class class class)co 7414 0cc0 11130 1c1 11131 + caddc 11133 < clt 11270 3c3 12290 4c4 12291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-2 12297 df-3 12298 df-4 12299 |
This theorem is referenced by: 4ne0 12342 5pos 12343 div4p1lem1div2 12489 fldiv4p1lem1div2 13824 iexpcyc 14194 discr 14226 faclbnd2 14274 sqrt2gt1lt2 15245 flodddiv4 16381 slotsdifplendx2 17389 pcoass 24938 csbren 25314 minveclem2 25341 dveflem 25898 sincos4thpi 26435 log2cnv 26863 chtublem 27131 bposlem6 27209 gausslemma2dlem0d 27279 2sqlem11 27349 chebbnd1lem3 27391 chebbnd1 27392 pntibndlem1 27509 pntlemb 27517 pntlemg 27518 pntlemr 27522 pntlemf 27525 usgrexmplef 29059 upgr4cycl4dv4e 29982 minvecolem2 30672 minvecolem3 30673 normlem6 30912 sqsscirc1 33445 hgt750lem 34219 iccioo01 36742 lcmineqlem23 41459 3lexlogpow2ineq2 41467 aks4d1p1p7 41482 aks4d1p1p5 41483 limclner 44962 stoweid 45374 stirlinglem10 45394 stirlinglem12 45396 bgoldbtbndlem3 47070 itsclc0yqsollem2 47759 itscnhlinecirc02plem1 47778 |
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