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Theorem onun2i 4377
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  e.  On
onun2i.2  |-  B  e.  On
Assertion
Ref Expression
onun2i  |-  ( A  u.  B )  e.  On

Proof of Theorem onun2i
StepHypRef Expression
1 onun2i.1 . 2  |-  A  e.  On
2 onun2i.2 . 2  |-  B  e.  On
3 onun2 4376 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  u.  B
)  e.  On )
41, 2, 3mp2an 422 1  |-  ( A  u.  B )  e.  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1465    u. cun 3039   Oncon0 4255
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260
This theorem is referenced by: (None)
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