![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nnoni | Structured version Visualization version GIF version |
Description: A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Ref | Expression |
---|---|
nnoni.1 | ⊢ 𝐴 ∈ ω |
Ref | Expression |
---|---|
nnoni | ⊢ 𝐴 ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnoni.1 | . 2 ⊢ 𝐴 ∈ ω | |
2 | nnon 7855 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Oncon0 6355 ωcom 7849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-in 3948 df-ss 3958 df-om 7850 |
This theorem is referenced by: omopthlem1 8655 omopthlem2 8656 omopthi 8657 |
Copyright terms: Public domain | W3C validator |