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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 7252 | . 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 2164 Vcvv 3414 ⊆ wss 3798 ∪ cuni 4660 Oncon0 5967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-tr 4978 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-ord 5970 df-on 5971 |
This theorem is referenced by: uniordint 7272 tz9.12lem2 8935 ttukeylem6 9658 onsetreclem2 43361 |
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