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Theorem onuni 6566
Description: The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
Assertion
Ref Expression
onuni  |-  ( A  e.  On  ->  U. A  e.  On )

Proof of Theorem onuni
StepHypRef Expression
1 onss 6564 . 2  |-  ( A  e.  On  ->  A  C_  On )
2 ssonuni 6560 . 2  |-  ( A  e.  On  ->  ( A  C_  On  ->  U. A  e.  On ) )
31, 2mpd 15 1  |-  ( A  e.  On  ->  U. A  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842    C_ wss 3413   U.cuni 4190   Oncon0 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825
This theorem is referenced by:  onuninsuci  6613  oeeulem  7207  cnfcom3lem  8099  cnfcom3lemOLD  8107  rankxpsuc  8252  dfac12lem2  8476  ttukeylem3  8843  r1limwun  9064  ontgval  30651  ordtoplem  30655  ordcmp  30667  aomclem1  35343
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