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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 8747 | . 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 2141 class class class wbr 3987 (class class class)co 5850 ℝcr 7760 1c1 7762 + caddc 7764 < clt 7941 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-iota 5158 df-fv 5204 df-ov 5853 df-pnf 7943 df-mnf 7944 df-ltxr 7946 |
This theorem is referenced by: zltp1le 9253 fznatpl1 10019 fzp1disj 10023 fzneuz 10044 fzp1nel 10047 fzonn0p1 10154 rebtwn2z 10198 seq3f1olemqsumk 10442 bernneq3 10585 bcp1nk 10683 bcpasc 10687 hashfzp1 10746 seq3coll 10764 resqrexlemover 10961 fsum1p 11368 cvgratnnlembern 11473 cvgratnnlemseq 11476 cvgratnnlemfm 11479 cvgratz 11482 mertenslemi1 11485 fprodntrivap 11534 fprod1p 11549 fprodeq0 11567 efcllemp 11608 nno 11852 zssinfcl 11890 sqrt2irr 12103 pcprendvds 12231 pcmpt 12282 1arith 12306 exmidunben 12368 nninfdclemp1 12392 suplociccreex 13355 cvgcmp2nlemabs 14024 |
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