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Mirrors > Home > ILE Home > Th. List > onsucb | GIF version |
Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4500. (Contributed by NM, 9-Sep-2003.) |
Ref | Expression |
---|---|
onsucb | ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsuc 4500 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
2 | eloni 4375 | . . 3 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴) | |
3 | elex 2748 | . . . . 5 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ∈ V) | |
4 | sucexb 4496 | . . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 134 | . . . 4 ⊢ (suc 𝐴 ∈ On → 𝐴 ∈ V) |
6 | elong 4373 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
7 | ordsucg 4501 | . . . . 5 ⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴)) | |
8 | 6, 7 | bitrd 188 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord suc 𝐴)) |
9 | 5, 8 | syl 14 | . . 3 ⊢ (suc 𝐴 ∈ On → (𝐴 ∈ On ↔ Ord suc 𝐴)) |
10 | 2, 9 | mpbird 167 | . 2 ⊢ (suc 𝐴 ∈ On → 𝐴 ∈ On) |
11 | 1, 10 | impbii 126 | 1 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 Vcvv 2737 Ord word 4362 Oncon0 4363 suc csuc 4365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-tr 4102 df-iord 4366 df-on 4368 df-suc 4371 |
This theorem is referenced by: onsucmin 4506 onsucuni2 4563 |
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