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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 10810 | . 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | nnon 7804 | . 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 6315 ωcom 7798 Ncnpi 10776 |
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 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-in 3915 df-ss 3925 df-om 7799 df-ni 10804 |
This theorem is referenced by: indpi 10839 nqereu 10861 |
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