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Mirrors > Home > ILE Home > Th. List > ordsucg | GIF version |
Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
Ref | Expression |
---|---|
ordsucg | ⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsucim 4424 | . 2 ⊢ (Ord 𝐴 → Ord suc 𝐴) | |
2 | sucidg 4346 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
3 | ordelord 4311 | . . . 4 ⊢ ((Ord suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → Ord 𝐴) | |
4 | 3 | ex 114 | . . 3 ⊢ (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴)) |
5 | 2, 4 | syl5com 29 | . 2 ⊢ (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴)) |
6 | 1, 5 | impbid2 142 | 1 ⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 1481 Vcvv 2689 Ord word 4292 suc csuc 4295 |
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 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-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 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-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-uni 3745 df-tr 4035 df-iord 4296 df-suc 4301 |
This theorem is referenced by: sucelon 4427 |
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