Theorem trin 4500

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 3608	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2		trss 4499	. . . . . 6 |- ( Tr A -> ( x e. A -> x C_ A ) )
3		trss 4499	. . . . . 6 |- ( Tr B -> ( x e. B -> x C_ B ) )
4	2, 3	im2anan9 853	. . . . 5 |- ( ( Tr A /\ Tr B ) -> ( ( x e. A /\ x e. B ) -> ( x C_ A /\ x C_ B ) ) )
5	1, 4	syl5bi 225	. . . 4 |- ( ( Tr A /\ Tr B ) -> ( x e. ( A i^i B ) -> ( x C_ A /\ x C_ B ) ) )
6		ssin 3645	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
7	5, 6	syl6ib 234	. . 3 |- ( ( Tr A /\ Tr B ) -> ( x e. ( A i^i B ) -> x C_ ( A i^i B ) ) )
8	7	ralrimiv 2808	. 2 |- ( ( Tr A /\ Tr B ) -> A. x e. ( A i^i B ) x C_ ( A i^i B ) )
9		dftr3 4494	. 2 |- ( Tr ( A i^i B ) <-> A. x e. ( A i^i B ) x C_ ( A i^i B ) )
10	8, 9	sylibr 217	1 |- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 376   e. wcel 1904   A.wral 2756   i^i cin 3389   C_ wss 3390   Tr wtr 4490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-in 3397  df-ss 3404  df-uni 4191  df-tr 4491

This theorem is referenced by:  ordin  5460  tcmin  8243  ingru  9258  gruina  9261  dfon2lem4  30503