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Mirrors > Home > MPE Home > Th. List > pion | Structured version Visualization version GIF version |
Description: A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pion | ⊢ (𝐴 ∈ N → 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 10921 | . 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | nnon 7882 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ N → 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Oncon0 6376 ωcom 7876 Ncnpi 10887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-ss 3964 df-om 7877 df-ni 10915 |
This theorem is referenced by: indpi 10950 nqereu 10972 |
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