Theorem onin 4371

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 4360	. . 3 |- ( A e. On -> Ord A )
2		eloni 4360	. . 3 |- ( B e. On -> Ord B )
3		ordin 4370	. . 3 |- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) )
4	1, 2, 3	syl2an 287	. 2 |- ( ( A e. On /\ B e. On ) -> Ord ( A i^i B ) )
5		simpl 108	. . 3 |- ( ( A e. On /\ B e. On ) -> A e. On )
6		inex1g 4125	. . 3 |- ( A e. On -> ( A i^i B ) e. _V )
7		elong 4358	. . 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 166	1 |- ( ( A e. On /\ B e. On ) -> ( A i^i B ) e. On )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   _Vcvv 2730   i^i cin 3120   Ord word 4347   Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  tfrlem5  6293