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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 7844 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Oncon0 6353 ωcom 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-in 3951 df-ss 3961 df-om 7839 |
This theorem is referenced by: omopthlem1 8641 omopthlem2 8642 omopthi 8643 |
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