Theorem trin 4156

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 3357	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2		trss 4155	. . . . . 6 |- ( Tr A -> ( x e. A -> x C_ A ) )
3		trss 4155	. . . . . 6 |- ( Tr B -> ( x e. B -> x C_ B ) )
4	2, 3	im2anan9 598	. . . . 5 |- ( ( Tr A /\ Tr B ) -> ( ( x e. A /\ x e. B ) -> ( x C_ A /\ x C_ B ) ) )
5	1, 4	biimtrid 152	. . . 4 |- ( ( Tr A /\ Tr B ) -> ( x e. ( A i^i B ) -> ( x C_ A /\ x C_ B ) ) )
6		ssin 3396	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
7	5, 6	imbitrdi 161	. . 3 |- ( ( Tr A /\ Tr B ) -> ( x e. ( A i^i B ) -> x C_ ( A i^i B ) ) )
8	7	ralrimiv 2579	. 2 |- ( ( Tr A /\ Tr B ) -> A. x e. ( A i^i B ) x C_ ( A i^i B ) )
9		dftr3 4150	. 2 |- ( Tr ( A i^i B ) <-> A. x e. ( A i^i B ) x C_ ( A i^i B ) )
10	8, 9	sylibr 134	1 |- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2177   A.wral 2485   i^i cin 3166   C_ wss 3167   Tr wtr 4146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-in 3173  df-ss 3180  df-uni 3853  df-tr 4147

This theorem is referenced by:  ordin  4436