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Theorem onuni 4775
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 4773 . 2  |-  ( A  e.  On  ->  A  C_  On )
2 ssonuni 4769 . 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 set class
Syntax hints:    -> wi 4    e. wcel 1726    C_ wss 3322   U.cuni 4017   Oncon0 4583
This theorem is referenced by:  onuninsuci  4822  oeeulem  6846  cnfcom3lem  7662  rankxpsuc  7808  dfac12lem2  8026  ttukeylem3  8393  r1limwun  8613  ontgval  26183  ordtoplem  26187  ordcmp  26199  aomclem1  27131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587
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