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Theorem onun2i 4475
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 4474 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
41, 2, 3mp2an 424 1 (𝐴𝐵) ∈ On
Colors of variables: wff set class
Syntax hints:  wcel 2141  cun 3119  Oncon0 4348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353
This theorem is referenced by: (None)
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