Theorem onin 4141

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 4130	. . 3 |- ( A e. On -> Ord A )
2		eloni 4130	. . 3 |- ( B e. On -> Ord B )
3		ordin 4140	. . 3 |- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) )
4	1, 2, 3	syl2an 283	. 2 |- ( ( A e. On /\ B e. On ) -> Ord ( A i^i B ) )
5		simpl 107	. . 3 |- ( ( A e. On /\ B e. On ) -> A e. On )
6		inex1g 3914	. . 3 |- ( A e. On -> ( A i^i B ) e. _V )
7		elong 4128	. . 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 165	1 |- ( ( A e. On /\ B e. On ) -> ( A i^i B ) e. On )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   _Vcvv 2601   i^i cin 2972   Ord word 4117   Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123

This theorem is referenced by:  tfrlem5  5953