Theorem trin 4536

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 3669	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2		trss 4535	. . . . . 6 |- ( Tr A -> ( x e. A -> x C_ A ) )
3		trss 4535	. . . . . 6 |- ( Tr B -> ( x e. B -> x C_ B ) )
4	2, 3	im2anan9 833	. . . . 5 |- ( ( Tr A /\ Tr B ) -> ( ( x e. A /\ x e. B ) -> ( x C_ A /\ x C_ B ) ) )
5	1, 4	syl5bi 217	. . . 4 |- ( ( Tr A /\ Tr B ) -> ( x e. ( A i^i B ) -> ( x C_ A /\ x C_ B ) ) )
6		ssin 3702	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
7	5, 6	syl6ib 226	. . 3 |- ( ( Tr A /\ Tr B ) -> ( x e. ( A i^i B ) -> x C_ ( A i^i B ) ) )
8	7	ralrimiv 2853	. 2 |- ( ( Tr A /\ Tr B ) -> A. x e. ( A i^i B ) x C_ ( A i^i B ) )
9		dftr3 4530	. 2 |- ( Tr ( A i^i B ) <-> A. x e. ( A i^i B ) x C_ ( A i^i B ) )
10	8, 9	sylibr 212	1 |- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1802   A.wral 2791   i^i cin 3457   C_ wss 3458   Tr wtr 4526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ral 2796  df-v 3095  df-in 3465  df-ss 3472  df-uni 4231  df-tr 4527

This theorem is referenced by:  ordin  4894  tcmin  8170  ingru  9191  gruina  9194  dfon2lem4  29186