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Mirrors > Home > MPE Home > Th. List > ssonunii | Structured version Visualization version GIF version |
Description: The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.) |
Ref | Expression |
---|---|
ssonuni.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ssonunii | ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssonuni.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ssonuni 7620 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ⊆ On → ∪ 𝐴 ∈ On)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2109 Vcvv 3430 ⊆ wss 3891 ∪ cuni 4844 Oncon0 6263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-11 2157 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-sb 2071 df-clab 2717 df-cleq 2731 df-clel 2817 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-tr 5196 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-ord 6266 df-on 6267 |
This theorem is referenced by: uniordint 7641 tz9.12lem2 9530 ttukeylem6 10254 onsetreclem2 46363 |
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