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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 10644 | . 2 ⊢ 1 ∈ ℝ | |
2 | 0lt1 11165 | . . 3 ⊢ 0 < 1 | |
3 | ltaddpos 11133 | . . 3 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 1 ↔ 𝐴 < (𝐴 + 1))) | |
4 | 2, 3 | mpbii 235 | . 2 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 < (𝐴 + 1)) |
5 | 1, 4 | mpan 688 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 class class class wbr 5069 (class class class)co 7159 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 < clt 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 |
This theorem is referenced by: lep1 11484 letrp1 11487 recp1lt1 11541 ledivp1 11545 ltp1i 11547 ltp1d 11573 sup2 11600 uzind 12077 ge0p1rp 12423 qbtwnxr 12596 xrsupsslem 12703 supxrunb1 12715 fzp1disj 12969 fzneuz 12991 fzp1nel 12994 fsequb 13346 caubnd 14721 rlim2lt 14857 o1fsum 15171 pcprendvds 16180 pcmpt 16231 iocopnst 23547 bndth 23565 ovolicc2lem3 24123 ioorcl2 24176 itg2const2 24345 reeff1olem 25037 axlowdimlem13 26743 icoreunrn 34644 poimirlem4 34900 poimirlem22 34918 mblfinlem1 34933 xrpnf 41768 limsupre3lem 42019 fourierdlem25 42424 smfresal 43070 |
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