Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10492 | . 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | nnon 7650 | . 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 2110 Oncon0 6213 ωcom 7644 Ncnpi 10458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-in 3873 df-ss 3883 df-om 7645 df-ni 10486 |
This theorem is referenced by: indpi 10521 nqereu 10543 |
Copyright terms: Public domain | W3C validator |