Theorem onin 5754

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 5733	. . 3 |- ( A e. On -> Ord A )
2		eloni 5733	. . 3 |- ( B e. On -> Ord B )
3		ordin 5753	. . 3 |- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) )
4	1, 2, 3	syl2an 494	. 2 |- ( ( A e. On /\ B e. On ) -> Ord ( A i^i B ) )
5		simpl 473	. . 3 |- ( ( A e. On /\ B e. On ) -> A e. On )
6		inex1g 4801	. . 3 |- ( A e. On -> ( A i^i B ) e. _V )
7		elong 5731	. . 3 |- ( ( A i^i B ) e. _V -> ( ( A i^i B ) e. On <-> Ord ( A i^i B ) ) )
8	5, 6, 7	3syl 18	. 2 |- ( ( A e. On /\ B e. On ) -> ( ( A i^i B ) e. On <-> Ord ( A i^i B ) ) )
9	4, 8	mpbird 247	1 |- ( ( A e. On /\ B e. On ) -> ( A i^i B ) e. On )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   i^i cin 3573   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-tr 4753  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  tfrlem5  7476  noreson  31813  ontopbas  32427