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Mirrors > Home > MPE Home > Th. List > peano2uz | Structured version Visualization version GIF version |
Description: Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.) |
Ref | Expression |
---|---|
peano2uz | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1081 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ) | |
2 | peano2z 11456 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
3 | 2 | 3ad2ant2 1103 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑁 + 1) ∈ ℤ) |
4 | zre 11419 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 11419 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | letrp1 10903 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) | |
7 | 5, 6 | syl3an2 1400 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) |
8 | 4, 7 | syl3an1 1399 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) |
9 | 1, 3, 8 | 3jca 1261 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 + 1))) |
10 | eluz2 11731 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
11 | eluz2 11731 | . 2 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 + 1))) | |
12 | 9, 10, 11 | 3imtr4i 281 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 1c1 9975 + caddc 9977 ≤ cle 10113 ℤcz 11415 ℤ≥cuz 11725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 |
This theorem is referenced by: peano2uzs 11780 peano2uzr 11781 uzaddcl 11782 fzsplit 12405 fzssp1 12422 fzsuc 12426 fzpred 12427 fzp1ss 12430 fzp1elp1 12432 fztp 12435 fzneuz 12459 fzosplitsnm1 12582 fzofzp1 12605 fzosplitsn 12616 fzosplitpr 12617 fzostep1 12624 om2uzuzi 12788 uzrdgsuci 12799 fzen2 12808 fzfi 12811 seqsplit 12874 seqf1olem1 12880 seqf1olem2 12881 seqz 12889 faclbnd3 13119 bcm1k 13142 seqcoll 13286 seqcoll2 13287 swrds1 13497 clim2ser 14429 clim2ser2 14430 serf0 14455 iseraltlem2 14457 iseralt 14459 fsump1 14531 fsump1i 14544 fsumparts 14582 cvgcmp 14592 isum1p 14617 isumsup2 14622 climcndslem1 14625 climcndslem2 14626 climcnds 14627 cvgrat 14659 mertenslem1 14660 clim2prod 14664 clim2div 14665 ntrivcvgfvn0 14675 fprodntriv 14716 fprodp1 14743 fprodabs 14748 binomfallfaclem2 14815 pcfac 15650 gsumprval 17328 telgsumfzslem 18431 telgsumfzs 18432 dvply2g 24085 aaliou3lem2 24143 ppinprm 24923 chtnprm 24925 ppiublem1 24972 chtublem 24981 chtub 24982 bposlem6 25059 pntlemf 25339 ostth2lem2 25368 clwwlkvbij 27088 clwwlkvbijOLD 27089 fzsplit3 29681 esumcvg 30276 sseqf 30582 gsumnunsn 30743 signstfvp 30776 iprodefisumlem 31752 poimirlem1 33540 poimirlem2 33541 poimirlem3 33542 poimirlem4 33543 poimirlem6 33545 poimirlem7 33546 poimirlem8 33547 poimirlem9 33548 poimirlem12 33551 poimirlem13 33552 poimirlem14 33553 poimirlem15 33554 poimirlem16 33555 poimirlem17 33556 poimirlem18 33557 poimirlem19 33558 poimirlem20 33559 poimirlem21 33560 poimirlem22 33561 poimirlem23 33562 poimirlem24 33563 poimirlem26 33565 poimirlem27 33566 poimirlem31 33570 poimirlem32 33571 sdclem2 33668 fdc 33671 mettrifi 33683 bfplem2 33752 rexrabdioph 37675 monotuz 37823 wallispilem1 40600 dirkertrigeqlem2 40634 sge0p1 40949 carageniuncllem1 41056 iccpartres 41679 iccelpart 41694 pfxccatpfx2 41753 fmtno4prm 41812 |
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