Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nnnn0d | GIF version |
Description: A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnnn0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nnnn0d | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssnn0 8184 | . 2 ⊢ ℕ ⊆ ℕ0 | |
2 | nnnn0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
3 | 1, 2 | sseldi 2943 | 1 ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ℕcn 7914 ℕ0cn0 8181 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-n0 8182 |
This theorem is referenced by: nn0ge2m1nn0 8243 nnzd 8359 eluzge2nn0 8512 expinnval 9258 expgt1 9293 expaddzaplem 9298 expaddzap 9299 expmulzap 9301 expnbnd 9372 resqrexlemnm 9616 resqrexlemcvg 9617 |
Copyright terms: Public domain | W3C validator |