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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 7798 | . 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
2 | uniexg 7759 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
3 | elong 6394 | . . 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 2106 Vcvv 3478 ⊆ wss 3963 ∪ cuni 4912 Ord word 6385 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: ssonunii 7800 onuni 7808 iunon 8378 onfununi 8380 oemapvali 9722 cardprclem 10017 carduni 10019 dfac12lem2 10183 ontgval 36414 onsupcl2 43214 onuniintrab 43215 onsupuni 43218 onsupcl3 43222 cantnfub2 43312 |
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