| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > nnssnn0 | Structured version Visualization version GIF version | ||
| Description: Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
| Ref | Expression |
|---|---|
| nnssnn0 | ⊢ ℕ ⊆ ℕ0 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssun1 4178 | . 2 ⊢ ℕ ⊆ (ℕ ∪ {0}) | |
| 2 | df-n0 12527 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
| 3 | 1, 2 | sseqtrri 4033 | 1 ⊢ ℕ ⊆ ℕ0 |
| Colors of variables: wff setvar class |
| Syntax hints: ∪ cun 3949 ⊆ wss 3951 {csn 4626 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-n0 12527 |
| This theorem is referenced by: nnnn0 12533 nnnn0d 12587 nthruz 16289 oddge22np1 16386 bitsfzolem 16471 lcmfval 16658 ramub1 17066 ramcl 17067 ply1divex 26176 pserdvlem2 26472 2sqreunnlem1 27493 2sqreunnlem2 27499 fsum2dsub 34622 breprexplemc 34647 breprexpnat 34649 knoppndvlem18 36530 sumcubes 42347 hbtlem5 43140 brfvtrcld 43734 corcltrcl 43752 fourierdlem50 46171 fourierdlem102 46223 fourierdlem114 46235 fmtnoinf 47523 fmtnofac2 47556 |
| Copyright terms: Public domain | W3C validator |