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Mirrors > Home > ILE Home > Th. List > pion | GIF version |
Description: A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) |
Ref | Expression |
---|---|
pion | ⊢ (𝐴 ∈ N → 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7110 | . 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | nnon 4518 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐴 ∈ N → 𝐴 ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 Oncon0 4280 ωcom 4499 Ncnpi 7073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-ni 7105 |
This theorem is referenced by: ltsopi 7121 indpi 7143 |
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