![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nn0negzi | GIF version |
Description: The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nn0negzi.1 | ⊢ 𝑁 ∈ ℕ0 |
Ref | Expression |
---|---|
nn0negzi | ⊢ -𝑁 ∈ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0negzi.1 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
2 | nn0negz 9337 | . 2 ⊢ (𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -𝑁 ∈ ℤ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 -cneg 8177 ℕ0cn0 9226 ℤcz 9303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-addcom 7958 ax-addass 7960 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-0id 7966 ax-rnegex 7967 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-ltadd 7974 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2758 df-sbc 2982 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-br 4026 df-opab 4087 df-id 4318 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-iota 5203 df-fun 5244 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-inn 8969 df-n0 9227 df-z 9304 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |