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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 4502. (Contributed by NM, 9-Sep-2003.) |
Ref | Expression |
---|---|
onsucb | ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsuc 4502 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
2 | eloni 4377 | . . 3 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴) | |
3 | elex 2750 | . . . . 5 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ∈ V) | |
4 | sucexb 4498 | . . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 134 | . . . 4 ⊢ (suc 𝐴 ∈ On → 𝐴 ∈ V) |
6 | elong 4375 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
7 | ordsucg 4503 | . . . . 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 2739 Ord word 4364 Oncon0 4365 suc csuc 4367 |
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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
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 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-uni 3812 df-tr 4104 df-iord 4368 df-on 4370 df-suc 4373 |
This theorem is referenced by: onsucmin 4508 onsucuni2 4565 |
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