Theorem onin 2978

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 2977	. . 3 |- ((Ord A /\ Ord B) -> Ord (A i^i B))
2		eloni 2958	. . 3 |- (A e. On -> Ord A)
3		eloni 2958	. . 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 2718	. . 3 |- (A e. On -> (A i^i B) e. V)
7		elong 2956	. . 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 958  Vcvv 1811   i^i cin 2046  Ord word 2947  Oncon0 2948

This theorem is referenced by:  tfrlem5 3915

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-tr 2681  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952

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