Theorem trin 2690

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 2689	. . . . . 6 |- (Tr A -> (x e. A -> x (_ A))
2		trss 2689	. . . . . 6 |- (Tr B -> (x e. B -> x (_ B))
3	1, 2	im2anan9 563	. . . . 5 |- ((Tr A /\ Tr B) -> ((x e. A /\ x e. B) -> (x (_ A /\ x (_ B)))
4		elin 2207	. . . . 5 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
5	3, 4	syl5ib 206	. . . 4 |- ((Tr A /\ Tr B) -> (x e. (A i^i B) -> (x (_ A /\ x (_ B)))
6		ssin 2232	. . . 4 |- ((x (_ A /\ x (_ B) <-> x (_ (A i^i B))
7	5, 6	syl6ib 212	. . 3 |- ((Tr A /\ Tr B) -> (x e. (A i^i B) -> x (_ (A i^i B)))
8	7	r19.21aiv 1713	. 2 |- ((Tr A /\ Tr B) -> A.x e. (A i^i B)x (_ (A i^i B))
9		dftr3 2684	. 2 |- (Tr (A i^i B) <-> A.x e. (A i^i B)x (_ (A i^i B))
10	8, 9	sylibr 200	1 |- ((Tr A /\ Tr B) -> Tr (A i^i B))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  A.wral 1645   i^i cin 2046   (_ wss 2047  Tr wtr 2680

This theorem is referenced by:  ordin 2977

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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504  df-tr 2681

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