Theorem trin 2051

Description: The intersection of transitive classes is transitive.

Assertion

Ref	Expression
trin	|- ((Tr A /\ Tr B) -> Tr (A i^i B))

Proof of Theorem trin

Step	Hyp	Ref	Expression
1		trss 2050	. . . . . 6 |- (Tr A -> (x e. A -> x (_ A))
2		trss 2050	. . . . . 6 |- (Tr B -> (x e. B -> x (_ B))
3	1, 2	im2anan9 434	. . . . 5 |- ((Tr A /\ Tr B) -> ((x e. A /\ x e. B) -> (x (_ A /\ x (_ B)))
4		elin 1635	. . . . 5 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
5	3, 4	syl5ib 181	. . . 4 |- ((Tr A /\ Tr B) -> (x e. (A i^i B) -> (x (_ A /\ x (_ B)))
6		ssin 1659	. . . 4 |- ((x (_ A /\ x (_ B) <-> x (_ (A i^i B))
7	5, 6	syl6ib 185	. . 3 |- ((Tr A /\ Tr B) -> (x e. (A i^i B) -> x (_ (A i^i B)))
8	7	r19.21aiv 1259	. 2 |- ((Tr A /\ Tr B) -> A.x e. (A i^i B)x (_ (A i^i B))
9		dftr3 2045	. 2 |- (Tr (A i^i B) <-> A.x e. (A i^i B)x (_ (A i^i B))
10	8, 9	sylibr 175	1 |- ((Tr A /\ Tr B) -> Tr (A i^i B))

Syntax hints:   -> wi 2   /\ wa 196   e. wcel 1092  A.wral 1201   i^i cin 1486   (_ wss 1487  Tr wtr 2041

This theorem is referenced by:  ordin 2228

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920  df-tr 2042

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