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Theorem onuni 4410
 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 4409 . 2 (𝐴 ∈ On → 𝐴 ⊆ On)
2 ssonuni 4404 . 2 (𝐴 ∈ On → (𝐴 ⊆ On → 𝐴 ∈ On))
31, 2mpd 13 1 (𝐴 ∈ On → 𝐴 ∈ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1480   ⊆ wss 3071  ∪ cuni 3736  Oncon0 4285 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290 This theorem is referenced by: (None)
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