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Theorem onuni 7371
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 7368 . 2 (𝐴 ∈ On → 𝐴 ⊆ On)
2 ssonuni 7364 . 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 2083  wss 3865   cuni 4751  Oncon0 6073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-tr 5071  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-ord 6076  df-on 6077
This theorem is referenced by:  onuninsuci  7418  oeeulem  8084  cnfcom3lem  9019  rankxpsuc  9164  dfac12lem2  9423  ttukeylem3  9786  r1limwun  10011  ontgval  33390  ordtoplem  33394  ordcmp  33406  1oequni2o  34201  rdgsucuni  34202  aomclem1  39160
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