Theorem trin 4555

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

Assertion

Ref	Expression
trin

Proof of Theorem trin

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3686	. . . . 5
2		trss 4554	. . . . . 6
3		trss 4554	. . . . . 6
4	2, 3	im2anan9 835	. . . . 5
5	1, 4	syl5bi 217	. . . 4
6		ssin 3719	. . . 4
7	5, 6	syl6ib 226	. . 3
8	7	ralrimiv 2869	. 2
9		dftr3 4549	. 2
10	8, 9	sylibr 212	1

Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  A.wral 2807  i^icin 3474  C_wss 3475  Trwtr 4545

This theorem is referenced by:  ordin  4913  tcmin  8193  ingru  9214  gruina  9217  dfon2lem4  29218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-in 3482  df-ss 3489  df-uni 4250  df-tr 4546