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Mirrors > Home > ILE Home > Th. List > ltp1d | GIF version |
Description: A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
ltp1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
ltp1d | ⊢ (𝜑 → 𝐴 < (𝐴 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltp1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltp1 8595 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝐴 < (𝐴 + 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℝcr 7612 1c1 7614 + caddc 7616 < clt 7793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-ltxr 7798 |
This theorem is referenced by: zltp1le 9101 fznatpl1 9849 fzp1disj 9853 fzneuz 9874 fzp1nel 9877 fzonn0p1 9981 rebtwn2z 10025 seq3f1olemqsumk 10265 bernneq3 10407 bcp1nk 10501 bcpasc 10505 hashfzp1 10563 seq3coll 10578 resqrexlemover 10775 fsum1p 11180 cvgratnnlembern 11285 cvgratnnlemseq 11288 cvgratnnlemfm 11291 cvgratz 11294 mertenslemi1 11297 efcllemp 11353 nno 11592 zssinfcl 11630 sqrt2irr 11829 exmidunben 11928 suplociccreex 12760 cvgcmp2nlemabs 13216 |
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