Theorem onin 2984

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 2983	. . 3 |- ((Ord A /\ Ord B) -> Ord (A i^i B))
2		eloni 2964	. . 3 |- (A e. On -> Ord A)
3		eloni 2964	. . 3 |- (B e. On -> Ord B)
4	1, 2, 3	syl2an 456	. 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 2723	. . 3 |- (A e. On -> (A i^i B) e. V)
7		elong 2962	. . 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 960  Vcvv 1814   i^i cin 2049  Ord word 2953  Oncon0 2954

This theorem is referenced by:  tfrlem5 3921

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-tr 2686  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958

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