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Mirrors > Home > ILE Home > Th. List > peano2uz | 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 986 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ) | |
2 | peano2z 9218 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
3 | 2 | 3ad2ant2 1008 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑁 + 1) ∈ ℤ) |
4 | zre 9186 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 9186 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | letrp1 8734 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) | |
7 | 5, 6 | syl3an2 1261 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) |
8 | 4, 7 | syl3an1 1260 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) |
9 | 1, 3, 8 | 3jca 1166 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 + 1))) |
10 | eluz2 9463 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
11 | eluz2 9463 | . 2 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 + 1))) | |
12 | 9, 10, 11 | 3imtr4i 200 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 967 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℝcr 7743 1c1 7745 + caddc 7747 ≤ cle 7925 ℤcz 9182 ℤ≥cuz 9457 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 |
This theorem is referenced by: peano2uzs 9513 peano2uzr 9514 uzaddcl 9515 fzsplit 9976 fzssp1 9992 fzsuc 9994 fzpred 9995 fzp1ss 9998 fzp1elp1 10000 fztp 10003 fzneuz 10026 fzosplitsnm1 10134 fzofzp1 10152 fzosplitsn 10158 fzostep1 10162 frec2uzuzd 10327 frecuzrdgrrn 10333 frec2uzrdg 10334 frecuzrdgrcl 10335 frecuzrdgsuc 10339 frecuzrdgrclt 10340 frecuzrdgg 10341 frecuzrdgsuctlem 10348 frecfzen2 10352 fzfig 10355 uzsinds 10367 iseqovex 10381 seq3val 10383 seqvalcd 10384 seqf 10386 seq3p1 10387 seq3split 10404 seq3homo 10435 seq3z 10436 ser3ge0 10442 faclbnd3 10645 bcm1k 10662 seq3coll 10741 clim2ser 11264 clim2ser2 11265 serf0 11279 fsump1 11347 fsump1i 11360 fsumparts 11397 isum1p 11419 cvgratnnlemmn 11452 mertenslemi1 11462 clim2prod 11466 clim2divap 11467 fprodntrivap 11511 fprodp1 11527 fprodabs 11543 zsupcllemstep 11863 infssuzex 11867 pcfac 12259 |
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