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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 4458 | . 2 ⊢ (Ord 𝐴 → Ord suc 𝐴) | |
2 | sucidg 4376 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
3 | ordelord 4341 | . . . 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 2128 Vcvv 2712 Ord word 4322 suc csuc 4325 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-uni 3773 df-tr 4063 df-iord 4326 df-suc 4331 |
This theorem is referenced by: sucelon 4461 |
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