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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 12359 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ) | |
2 | 1 | 3ad2ant1 1132 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑀 + 1) ∈ ℤ) |
3 | peano2z 12359 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
4 | 3 | 3ad2ant2 1133 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑁 + 1) ∈ ℤ) |
5 | zre 12321 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
6 | zre 12321 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
7 | 1re 10973 | . . . . . 6 ⊢ 1 ∈ ℝ | |
8 | leadd1 11441 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) | |
9 | 7, 8 | mp3an3 1449 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
10 | 5, 6, 9 | syl2an 596 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
11 | 10 | biimp3a 1468 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑀 + 1) ≤ (𝑁 + 1)) |
12 | 2, 4, 11 | 3jca 1127 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → ((𝑀 + 1) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ (𝑀 + 1) ≤ (𝑁 + 1))) |
13 | eluz2 12586 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
14 | eluz2 12586 | . 2 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ↔ ((𝑀 + 1) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ (𝑀 + 1) ≤ (𝑁 + 1))) | |
15 | 12, 13, 14 | 3imtr4i 292 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5076 ‘cfv 6435 (class class class)co 7277 ℝcr 10868 1c1 10870 + caddc 10872 ≤ cle 11008 ℤcz 12317 ℤ≥cuz 12580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-n0 12232 df-z 12318 df-uz 12581 |
This theorem is referenced by: uzp1 12617 fzp1elp1 13307 seqcl2 13739 seqfveq2 13743 seqf1olem2 13761 seqid2 13767 seqcoll 14176 serf0 15390 efcllem 15785 prmind2 16388 pockthlem 16604 pockthg 16605 prmunb 16613 prmreclem4 16618 dvradcnv 25578 rplogsumlem1 26630 rplogsumlem2 26631 dchrisumlem2 26636 dchrisum0flb 26656 pntlemq 26747 pntlemr 26748 pntlemf 26751 axlowdimlem17 27324 fibp1 32365 subfacp1lem5 33143 poimirlem1 35775 poimirlem3 35777 poimirlem4 35778 poimirlem15 35789 poimirlem16 35790 poimirlem17 35791 poimirlem19 35793 poimirlem20 35794 poimirlem23 35797 fdc 35900 mettrifi 35912 expdiophlem1 40840 trclfvdecomr 41306 |
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