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Mirrors > Home > ILE Home > Th. List > elon2 | GIF version |
Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.) |
Ref | Expression |
---|---|
elon2 | ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4393 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | elex 2763 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V) | |
3 | 1, 2 | jca 306 | . 2 ⊢ (𝐴 ∈ On → (Ord 𝐴 ∧ 𝐴 ∈ V)) |
4 | elong 4391 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
5 | 4 | biimparc 299 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ On) |
6 | 3, 5 | impbii 126 | 1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2160 Vcvv 2752 Ord word 4380 Oncon0 4381 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-in 3150 df-ss 3157 df-uni 3825 df-tr 4117 df-iord 4384 df-on 4386 |
This theorem is referenced by: tfrexlem 6358 pw1on 7254 |
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