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Mirrors > Home > MPE Home > Th. List > eluzp1p1 | Structured version Visualization version GIF version |
Description: Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.) |
Ref | Expression |
---|---|
eluzp1p1 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2z 11775 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ) | |
2 | 1 | 3ad2ant1 1124 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑀 + 1) ∈ ℤ) |
3 | peano2z 11775 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
4 | 3 | 3ad2ant2 1125 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑁 + 1) ∈ ℤ) |
5 | zre 11737 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
6 | zre 11737 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
7 | 1re 10378 | . . . . . 6 ⊢ 1 ∈ ℝ | |
8 | leadd1 10846 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) | |
9 | 7, 8 | mp3an3 1523 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
10 | 5, 6, 9 | syl2an 589 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
11 | 10 | biimp3a 1542 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑀 + 1) ≤ (𝑁 + 1)) |
12 | 2, 4, 11 | 3jca 1119 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → ((𝑀 + 1) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ (𝑀 + 1) ≤ (𝑁 + 1))) |
13 | eluz2 12003 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
14 | eluz2 12003 | . 2 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ↔ ((𝑀 + 1) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ (𝑀 + 1) ≤ (𝑁 + 1))) | |
15 | 12, 13, 14 | 3imtr4i 284 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1071 ∈ wcel 2107 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 ℝcr 10273 1c1 10275 + caddc 10277 ≤ cle 10414 ℤcz 11733 ℤ≥cuz 11997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-n0 11648 df-z 11734 df-uz 11998 |
This theorem is referenced by: uzp1 12032 fzp1elp1 12716 seqcl2 13142 seqfveq2 13146 seqf1olem2 13164 seqid2 13170 seqcoll 13568 serf0 14828 efcllem 15219 prmind2 15814 pockthlem 16024 pockthg 16025 prmunb 16033 prmreclem4 16038 dvradcnv 24623 rplogsumlem1 25642 rplogsumlem2 25643 dchrisumlem2 25648 dchrisum0flb 25668 pntlemq 25759 pntlemr 25760 pntlemf 25763 axlowdimlem17 26324 fibp1 31070 subfacp1lem5 31773 poimirlem1 34045 poimirlem3 34047 poimirlem4 34048 poimirlem15 34059 poimirlem16 34060 poimirlem17 34061 poimirlem19 34063 poimirlem20 34064 poimirlem23 34067 fdc 34174 mettrifi 34186 expdiophlem1 38561 trclfvdecomr 38991 |
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