Theorem onun2i 4539

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 4538	. 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 2176   u. cun 3164   Oncon0 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415

This theorem is referenced by: (None)