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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 10916 | . 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | nnon 7893 | . 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 2106 Oncon0 6386 ωcom 7887 Ncnpi 10882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-ss 3980 df-om 7888 df-ni 10910 |
This theorem is referenced by: indpi 10945 nqereu 10967 |
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