Theorem onun2i 4582

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 4581	. 2 |- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On )
4	1, 2, 3	mp2an 426	1 |- ( A u. B ) e. On

Syntax hints:   e. wcel 2200   u. cun 3195   Oncon0 4453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-uni 3888  df-tr 4182  df-iord 4456  df-on 4458

This theorem is referenced by: (None)