Theorem onin 4526

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 4515	. . 3 |- ( A e. On -> Ord A )
2		eloni 4515	. . 3 |- ( B e. On -> Ord B )
3		ordin 4525	. . 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 4264	. . 3 |- ( A e. On -> ( A i^i B ) e. _V )
7		elong 4513	. . 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 2209   _Vcvv 2821   i^i cin 3219   Ord word 4502   Oncon0 4503

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508

This theorem is referenced by:  tfrlem5  6575