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Theorem onun2i 4345
 Description: The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.)
Hypotheses
Ref Expression
onun2i.1 𝐴 ∈ On
onun2i.2 𝐵 ∈ On
Assertion
Ref Expression
onun2i (𝐴𝐵) ∈ On

Proof of Theorem onun2i
StepHypRef Expression
1 onun2i.1 . 2 𝐴 ∈ On
2 onun2i.2 . 2 𝐵 ∈ On
3 onun2 4344 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
41, 2, 3mp2an 420 1 (𝐴𝐵) ∈ On
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1448   ∪ cun 3019  Oncon0 4223 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pr 4069  ax-un 4293 This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-uni 3684  df-tr 3967  df-iord 4226  df-on 4228 This theorem is referenced by: (None)
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