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Mirrors > Home > ILE Home > Th. List > suceloni | GIF version |
Description: The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) |
Ref | Expression |
---|---|
suceloni | ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4267 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordsucim 4386 | . . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
4 | sucexg 4384 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V) | |
5 | elong 4265 | . . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴)) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐴 ∈ On → (suc 𝐴 ∈ On ↔ Ord suc 𝐴)) |
7 | 3, 6 | mpbird 166 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 1465 Vcvv 2660 Ord word 4254 Oncon0 4255 suc csuc 4257 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 |
This theorem is referenced by: sucelon 4389 unon 4397 onsuci 4402 ordsucunielexmid 4416 tfrlemisucaccv 6190 tfrexlem 6199 tfri1dALT 6216 rdgisuc1 6249 rdgon 6251 oacl 6324 oasuc 6328 omsuc 6336 |
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