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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 7490 | . 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 2105 Vcvv 3492 ⊆ wss 3933 ∪ cuni 4830 Oncon0 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 |
This theorem is referenced by: uniordint 7510 tz9.12lem2 9205 ttukeylem6 9924 onsetreclem2 44736 |
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