Theorem onin 4417

Description: The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)

Assertion

Ref	Expression
onin	|- ( ( A e. On /\ B e. On ) -> ( A i^i B ) e. On )

Proof of Theorem onin

Step	Hyp	Ref	Expression
1		eloni 4406	. . 3 |- ( A e. On -> Ord A )
2		eloni 4406	. . 3 |- ( B e. On -> Ord B )
3		ordin 4416	. . 3 |- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) )
4	1, 2, 3	syl2an 289	. 2 |- ( ( A e. On /\ B e. On ) -> Ord ( A i^i B ) )
5		simpl 109	. . 3 |- ( ( A e. On /\ B e. On ) -> A e. On )
6		inex1g 4165	. . 3 |- ( A e. On -> ( A i^i B ) e. _V )
7		elong 4404	. . 3 |- ( ( A i^i B ) e. _V -> ( ( A i^i B ) e. On <-> Ord ( A i^i B ) ) )
8	5, 6, 7	3syl 17	. 2 |- ( ( A e. On /\ B e. On ) -> ( ( A i^i B ) e. On <-> Ord ( A i^i B ) ) )
9	4, 8	mpbird 167	1 |- ( ( A e. On /\ B e. On ) -> ( A i^i B ) e. On )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2164   _Vcvv 2760   i^i cin 3152   Ord word 4393   Oncon0 4394

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-sep 4147

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399

This theorem is referenced by:  tfrlem5  6367