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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 1054 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ) | |
2 | peano2z 11295 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
3 | 2 | 3ad2ant2 1076 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑁 + 1) ∈ ℤ) |
4 | zre 11258 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 11258 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | letrp1 10744 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) | |
7 | 5, 6 | syl3an2 1352 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) |
8 | 4, 7 | syl3an1 1351 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) |
9 | 1, 3, 8 | 3jca 1235 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 + 1))) |
10 | eluz2 11569 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
11 | eluz2 11569 | . 2 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 + 1))) | |
12 | 9, 10, 11 | 3imtr4i 280 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 1c1 9816 + caddc 9818 ≤ cle 9954 ℤcz 11254 ℤ≥cuz 11563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 |
This theorem is referenced by: peano2uzs 11618 peano2uzr 11619 uzaddcl 11620 fzsplit 12238 fzssp1 12255 fzsuc 12258 fzpred 12259 fzp1ss 12262 fzp1elp1 12264 fztp 12267 fzneuz 12290 fzosplitsnm1 12409 fzofzp1 12431 fzosplitsn 12442 fzostep1 12446 om2uzuzi 12610 uzrdgsuci 12621 fzen2 12630 fzfi 12633 seqsplit 12696 seqf1olem1 12702 seqf1olem2 12703 seqz 12711 faclbnd3 12941 bcm1k 12964 seqcoll 13105 seqcoll2 13106 swrds1 13303 clim2ser 14233 clim2ser2 14234 serf0 14259 iseraltlem2 14261 iseralt 14263 fsump1 14329 fsump1i 14342 fsumparts 14379 cvgcmp 14389 isum1p 14412 isumsup2 14417 climcndslem1 14420 climcndslem2 14421 climcnds 14422 cvgrat 14454 mertenslem1 14455 clim2prod 14459 clim2div 14460 ntrivcvgfvn0 14470 fprodntriv 14511 fprodp1 14538 fprodabs 14543 binomfallfaclem2 14610 pcfac 15441 gsumprval 17104 telgsumfzslem 18208 telgsumfzs 18209 dvply2g 23844 aaliou3lem2 23902 ppinprm 24678 chtnprm 24680 ppiublem1 24727 chtublem 24736 chtub 24737 bposlem6 24814 pntlemf 25094 ostth2lem2 25123 clwwlkvbij 26329 fzsplit3 28940 esumcvg 29475 sseqf 29781 gsumnunsn 29942 signstfvp 29974 iprodefisumlem 30879 poimirlem1 32580 poimirlem2 32581 poimirlem3 32582 poimirlem4 32583 poimirlem6 32585 poimirlem7 32586 poimirlem8 32587 poimirlem9 32588 poimirlem12 32591 poimirlem13 32592 poimirlem14 32593 poimirlem15 32594 poimirlem16 32595 poimirlem17 32596 poimirlem18 32597 poimirlem19 32598 poimirlem20 32599 poimirlem21 32600 poimirlem22 32601 poimirlem23 32602 poimirlem24 32603 poimirlem26 32605 poimirlem27 32606 poimirlem31 32610 poimirlem32 32611 sdclem2 32708 fdc 32711 mettrifi 32723 bfplem2 32792 rexrabdioph 36376 monotuz 36524 wallispilem1 38958 dirkertrigeqlem2 38992 sge0p1 39307 carageniuncllem1 39411 iccpartres 39956 iccelpart 39971 fmtno4prm 40025 pfxccatpfx2 40291 fzosplitpr 40362 clwwlksvbij 41229 |
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