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Theorem onuni 7227
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 7224 . 2 (𝐴 ∈ On → 𝐴 ⊆ On)
2 ssonuni 7220 . 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 2157  wss 3769   cuni 4628  Oncon0 5941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944  df-on 5945
This theorem is referenced by:  onuninsuci  7274  oeeulem  7921  cnfcom3lem  8850  rankxpsuc  8995  dfac12lem2  9254  ttukeylem3  9621  r1limwun  9846  ontgval  32938  ordtoplem  32942  ordcmp  32954  1oequni2o  33714  rdgsucuni  33715  aomclem1  38409
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