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Mirrors > Home > ILE Home > Th. List > onelon | GIF version |
Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.) |
Ref | Expression |
---|---|
onelon | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4257 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordelon 4265 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | |
3 | 1, 2 | sylan 279 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1463 Ord word 4244 Oncon0 4245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-in 3043 df-ss 3050 df-uni 3703 df-tr 3987 df-iord 4248 df-on 4250 |
This theorem is referenced by: oneli 4310 ssorduni 4363 unon 4387 tfrlemibacc 6177 tfrlemibxssdm 6178 tfrlemibfn 6179 tfrexlem 6185 tfr1onlemsucaccv 6192 tfrcllemsucaccv 6205 sucinc2 6296 oav2 6313 omv2 6315 |
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