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Mirrors > Home > MPE Home > Th. List > ltp1 | Structured version Visualization version GIF version |
Description: A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
Ref | Expression |
---|---|
ltp1 | ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 11209 | . 2 ⊢ 1 ∈ ℝ | |
2 | 0lt1 11731 | . . 3 ⊢ 0 < 1 | |
3 | ltaddpos 11699 | . . 3 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 1 ↔ 𝐴 < (𝐴 + 1))) | |
4 | 2, 3 | mpbii 232 | . 2 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 < (𝐴 + 1)) |
5 | 1, 4 | mpan 689 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 class class class wbr 5146 (class class class)co 7403 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 < clt 11243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 |
This theorem is referenced by: lep1 12050 letrp1 12053 recp1lt1 12107 ledivp1 12111 ltp1i 12113 ltp1d 12139 sup2 12165 uzind 12649 ge0p1rp 13000 qbtwnxr 13174 xrsupsslem 13281 supxrunb1 13293 fzp1disj 13555 fzneuz 13577 fzp1nel 13580 fsequb 13935 caubnd 15300 rlim2lt 15436 o1fsum 15754 pcprendvds 16768 pcmpt 16820 iocopnst 24437 bndth 24455 ovolicc2lem3 25017 ioorcl2 25070 itg2const2 25240 reeff1olem 25939 axlowdimlem13 28191 icoreunrn 36177 poimirlem4 36429 poimirlem22 36447 mblfinlem1 36462 aks4d1p1p4 40873 xrpnf 44130 limsupre3lem 44382 fourierdlem25 44782 smfresal 45438 natlocalincr 45524 |
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