Theorem onin 2968

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 2967	. . 3 |- ((Ord A /\ Ord B) -> Ord (A i^i B))
2		eloni 2948	. . 3 |- (A e. On -> Ord A)
3		eloni 2948	. . 3 |- (B e. On -> Ord B)
4	1, 2, 3	syl2an 454	. 2 |- ((A e. On /\ B e. On) -> Ord (A i^i B))
5		pm3.26 319	. . 3 |- ((A e. On /\ B e. On) -> A e. On)
6		inex1g 2708	. . 3 |- (A e. On -> (A i^i B) e. V)
7		elong 2946	. . 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 196	1 |- ((A e. On /\ B e. On) -> (A i^i B) e. On)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 955  Vcvv 1802   i^i cin 2036  Ord word 2937  Oncon0 2938

This theorem is referenced by:  tfrlem5 3900

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-tr 2671  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942

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