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Theorem onuni 7638
Description: The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
Assertion
Ref Expression
onuni (𝐴 ∈ On → 𝐴 ∈ On)

Proof of Theorem onuni
StepHypRef Expression
1 onss 7634 . 2 (𝐴 ∈ On → 𝐴 ⊆ On)
2 ssonuni 7630 . 2 (𝐴 ∈ On → (𝐴 ⊆ On → 𝐴 ∈ On))
31, 2mpd 15 1 (𝐴 ∈ On → 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wss 3887   cuni 4839  Oncon0 6266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270
This theorem is referenced by:  onuninsuci  7687  oeeulem  8432  cnfcom3lem  9461  rankxpsuc  9640  dfac12lem2  9900  ttukeylem3  10267  r1limwun  10492  ontgval  34620  ordtoplem  34624  ordcmp  34636  1oequni2o  35539  rdgsucuni  35540  aomclem1  40879
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