Theorem onin 3035

Description: The intersection of two ordinal numbers is an ordinal number.

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		ordin 3034	. . 3 |- ((Ord A /\ Ord B) -> Ord (A i^i B))
2		eloni 3015	. . 3 |- (A e. On -> Ord A)
3		eloni 3015	. . 3 |- (B e. On -> Ord B)
4	1, 2, 3	syl2an 465	. 2 |- ((A e. On /\ B e. On) -> Ord (A i^i B))
5		pm3.26 326	. . 3 |- ((A e. On /\ B e. On) -> A e. On)
6		inex1g 2773	. . 3 |- (A e. On -> (A i^i B) e. V)
7		elong 3013	. . 3 |- ((A i^i B) e. V -> ((A i^i B) e. On <-> Ord (A i^i B)))
8	5, 6, 7	3syl 20	. 2 |- ((A e. On /\ B e. On) -> ((A i^i B) e. On <-> Ord (A i^i B)))
9	4, 8	mpbird 203	1 |- ((A e. On /\ B e. On) -> (A i^i B) e. On)

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230   e. wcel 999  Vcvv 1858   i^i cin 2097  Ord word 3004  Oncon0 3005

This theorem is referenced by:  tfrlem5 3973

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-ral 1696  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-br 2675  df-tr 2736  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009

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