Theorem trin 4763

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 3796	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2		trss 4761	. . . . . 6 |- ( Tr A -> ( x e. A -> x C_ A ) )
3		trss 4761	. . . . . 6 |- ( Tr B -> ( x e. B -> x C_ B ) )
4	2, 3	im2anan9 880	. . . . 5 |- ( ( Tr A /\ Tr B ) -> ( ( x e. A /\ x e. B ) -> ( x C_ A /\ x C_ B ) ) )
5	1, 4	syl5bi 232	. . . 4 |- ( ( Tr A /\ Tr B ) -> ( x e. ( A i^i B ) -> ( x C_ A /\ x C_ B ) ) )
6		ssin 3835	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
7	5, 6	syl6ib 241	. . 3 |- ( ( Tr A /\ Tr B ) -> ( x e. ( A i^i B ) -> x C_ ( A i^i B ) ) )
8	7	ralrimiv 2965	. 2 |- ( ( Tr A /\ Tr B ) -> A. x e. ( A i^i B ) x C_ ( A i^i B ) )
9		dftr3 4756	. 2 |- ( Tr ( A i^i B ) <-> A. x e. ( A i^i B ) x C_ ( A i^i B ) )
10	8, 9	sylibr 224	1 |- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-tr 4753

This theorem is referenced by:  ordin  5753  tcmin  8617  ingru  9637  gruina  9640  dfon2lem4  31691