![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0xnn0 | Structured version Visualization version GIF version |
Description: Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
0xnn0 | ⊢ 0 ∈ ℕ0* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssxnn0 12542 | . 2 ⊢ ℕ0 ⊆ ℕ0* | |
2 | 0nn0 12482 | . 2 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | sselii 3977 | 1 ⊢ 0 ∈ ℕ0* |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 0cc0 11105 ℕ0cn0 12467 ℕ0*cxnn0 12539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-mulcl 11167 ax-i2m1 11173 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-un 3951 df-in 3953 df-ss 3963 df-sn 4627 df-n0 12468 df-xnn0 12540 |
This theorem is referenced by: 0edg0rgr 28808 rgrusgrprc 28825 rusgrprc 28826 rgrprcx 28828 |
Copyright terms: Public domain | W3C validator |