Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 992 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ) | |
2 | peano2z 9248 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
3 | 2 | 3ad2ant2 1014 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑁 + 1) ∈ ℤ) |
4 | zre 9216 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 9216 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | letrp1 8764 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) | |
7 | 5, 6 | syl3an2 1267 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) |
8 | 4, 7 | syl3an1 1266 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) |
9 | 1, 3, 8 | 3jca 1172 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 + 1))) |
10 | eluz2 9493 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
11 | eluz2 9493 | . 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 973 ∈ wcel 2141 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 ℝcr 7773 1c1 7775 + caddc 7777 ≤ cle 7955 ℤcz 9212 ℤ≥cuz 9487 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 |
This theorem is referenced by: peano2uzs 9543 peano2uzr 9544 uzaddcl 9545 fzsplit 10007 fzssp1 10023 fzsuc 10025 fzpred 10026 fzp1ss 10029 fzp1elp1 10031 fztp 10034 fzneuz 10057 fzosplitsnm1 10165 fzofzp1 10183 fzosplitsn 10189 fzostep1 10193 frec2uzuzd 10358 frecuzrdgrrn 10364 frec2uzrdg 10365 frecuzrdgrcl 10366 frecuzrdgsuc 10370 frecuzrdgrclt 10371 frecuzrdgg 10372 frecuzrdgsuctlem 10379 frecfzen2 10383 fzfig 10386 uzsinds 10398 iseqovex 10412 seq3val 10414 seqvalcd 10415 seqf 10417 seq3p1 10418 seq3split 10435 seq3homo 10466 seq3z 10467 ser3ge0 10473 faclbnd3 10677 bcm1k 10694 seq3coll 10777 clim2ser 11300 clim2ser2 11301 serf0 11315 fsump1 11383 fsump1i 11396 fsumparts 11433 isum1p 11455 cvgratnnlemmn 11488 mertenslemi1 11498 clim2prod 11502 clim2divap 11503 fprodntrivap 11547 fprodp1 11563 fprodabs 11579 zsupcllemstep 11900 infssuzex 11904 pcfac 12302 |
Copyright terms: Public domain | W3C validator |