Theorem trin 4097

Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)

Assertion

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

Proof of Theorem trin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3310	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2		trss 4096	. . . . . 6 |- ( Tr A -> ( x e. A -> x C_ A ) )
3		trss 4096	. . . . . 6 |- ( Tr B -> ( x e. B -> x C_ B ) )
4	2, 3	im2anan9 593	. . . . 5 |- ( ( Tr A /\ Tr B ) -> ( ( x e. A /\ x e. B ) -> ( x C_ A /\ x C_ B ) ) )
5	1, 4	syl5bi 151	. . . 4 |- ( ( Tr A /\ Tr B ) -> ( x e. ( A i^i B ) -> ( x C_ A /\ x C_ B ) ) )
6		ssin 3349	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
7	5, 6	syl6ib 160	. . 3 |- ( ( Tr A /\ Tr B ) -> ( x e. ( A i^i B ) -> x C_ ( A i^i B ) ) )
8	7	ralrimiv 2542	. 2 |- ( ( Tr A /\ Tr B ) -> A. x e. ( A i^i B ) x C_ ( A i^i B ) )
9		dftr3 4091	. 2 |- ( Tr ( A i^i B ) <-> A. x e. ( A i^i B ) x C_ ( A i^i B ) )
10	8, 9	sylibr 133	1 |- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   A.wral 2448   i^i cin 3120   C_ wss 3121   Tr wtr 4087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088

This theorem is referenced by:  ordin  4370