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Theorem onuni 7527
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 7524 . 2 (𝐴 ∈ On → 𝐴 ⊆ On)
2 ssonuni 7520 . 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 2114  wss 3843   cuni 4796  Oncon0 6172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-11 2162  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-ord 6175  df-on 6176
This theorem is referenced by:  onuninsuci  7574  oeeulem  8258  cnfcom3lem  9239  rankxpsuc  9384  dfac12lem2  9644  ttukeylem3  10011  r1limwun  10236  ontgval  34258  ordtoplem  34262  ordcmp  34274  1oequni2o  35162  rdgsucuni  35163  aomclem1  40451
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