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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 4112 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordsucim 4226 | . . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
4 | sucexg 4224 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V) | |
5 | elong 4110 | . . 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 156 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∈ wcel 1393 Vcvv 2557 Ord word 4099 Oncon0 4100 suc csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 |
This theorem is referenced by: sucelon 4229 unon 4237 onsuci 4242 ordsucunielexmid 4256 tfrlemisucaccv 5939 tfrexlem 5948 rdgisuc1 5971 frecsuclemdm 5988 oacl 6040 oasuc 6044 omsuc 6051 |
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