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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 10487 | . 2 ⊢ 1 ∈ ℝ | |
2 | 0lt1 11010 | . . 3 ⊢ 0 < 1 | |
3 | ltaddpos 10978 | . . 3 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 1 ↔ 𝐴 < (𝐴 + 1))) | |
4 | 2, 3 | mpbii 234 | . 2 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 < (𝐴 + 1)) |
5 | 1, 4 | mpan 686 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2081 class class class wbr 4962 (class class class)co 7016 ℝcr 10382 0cc0 10383 1c1 10384 + caddc 10386 < clt 10521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-po 5362 df-so 5363 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 |
This theorem is referenced by: lep1 11329 letrp1 11332 recp1lt1 11386 ledivp1 11390 ltp1i 11392 ltp1d 11418 sup2 11445 uzind 11923 ge0p1rp 12270 qbtwnxr 12443 xrsupsslem 12550 supxrunb1 12562 fzp1disj 12816 fzneuz 12838 fzp1nel 12841 fsequb 13193 caubnd 14552 rlim2lt 14688 o1fsum 15001 pcprendvds 16006 pcmpt 16057 iocopnst 23227 bndth 23245 ovolicc2lem3 23803 ioorcl2 23856 itg2const2 24025 reeff1olem 24717 axlowdimlem13 26423 icoreunrn 34171 poimirlem4 34427 poimirlem22 34445 mblfinlem1 34460 xrpnf 41304 limsupre3lem 41555 fourierdlem25 41959 smfresal 42605 |
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