Theorem onun2i 4557

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 4556	. 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 2178   u. cun 3172   Oncon0 4428

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433

This theorem is referenced by: (None)