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| Mirrors > Home > MPE Home > Th. List > ssonuni | Structured version Visualization version GIF version | ||
| Description: The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. Lemma 2.7 of [Schloeder] p. 4. (Contributed by NM, 1-Nov-2003.) |
| Ref | Expression |
|---|---|
| ssonuni | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssorduni 7799 | . 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
| 2 | uniexg 7760 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
| 3 | elong 6392 | . . 3 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴)) | |
| 4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴)) |
| 5 | 1, 4 | imbitrrid 246 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∪ cuni 4907 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: ssonunii 7801 onuni 7808 iunon 8379 onfununi 8381 oemapvali 9724 cardprclem 10019 carduni 10021 dfac12lem2 10185 ontgval 36432 onsupcl2 43237 onuniintrab 43238 onsupuni 43241 onsupcl3 43245 cantnfub2 43335 |
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