Theorem onun2i 4467

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

Step	Hyp	Ref	Expression
1		onun2i.1	. 2 |- A e. On
2		onun2i.2	. 2 |- B e. On
3		onun2 4466	. 2 |- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On )
4	1, 2, 3	mp2an 423	1 |- ( A u. B ) e. On

Syntax hints:   e. wcel 2136   u. cun 3113   Oncon0 4340

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pr 4186  ax-un 4410

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-sn 3581  df-pr 3582  df-uni 3789  df-tr 4080  df-iord 4343  df-on 4345

This theorem is referenced by: (None)