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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nnelon | GIF version |
Description: A natural number is an ordinal. Constructive proof of nnon 4531. Can also be proved from bj-omssonALT 13332. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnelon | ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnord 13327 | . 2 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | elong 4303 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ On ↔ Ord 𝐴)) | |
3 | 1, 2 | mpbird 166 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 Ord word 4292 Oncon0 4293 ωcom 4512 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-nul 4062 ax-pr 4139 ax-un 4363 ax-bd0 13182 ax-bdor 13185 ax-bdal 13187 ax-bdex 13188 ax-bdeq 13189 ax-bdel 13190 ax-bdsb 13191 ax-bdsep 13253 ax-infvn 13310 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-bdc 13210 df-bj-ind 13296 |
This theorem is referenced by: bj-omsson 13331 |
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