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Mirrors > Home > ILE Home > Th. List > ssonuni | GIF version |
Description: The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.) |
Ref | Expression |
---|---|
ssonuni | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssorduni 4398 | . 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
2 | uniexg 4356 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
3 | elong 4290 | . . 3 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴)) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴)) |
5 | 1, 4 | syl5ibr 155 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 ∪ cuni 3731 Ord word 4279 Oncon0 4280 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-in 3072 df-ss 3079 df-uni 3732 df-tr 4022 df-iord 4283 df-on 4285 |
This theorem is referenced by: ssonunii 4400 onun2 4401 onuni 4405 iunon 6174 |
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