Theorem trint 4560

Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)

Assertion

Ref	Expression
trint

Distinct variable group:   ,

Proof of Theorem trint

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr3 4549	. . . . 5
2	1	ralbii 2888	. . . 4
3		df-ral 2812	. . . . . 6
4	3	ralbii 2888	. . . . 5
5		ralcom4 3128	. . . . 5
6	4, 5	bitri 249	. . . 4
7	2, 6	sylbb 197	. . 3
8		ralim 2846	. . . 4
9	8	alimi 1633	. . 3
10	7, 9	syl 16	. 2
11		dftr3 4549	. . 3
12		df-ral 2812	. . . 4
13		vex 3112	. . . . . . 7
14	13	elint2 4293	. . . . . 6
15		ssint 4302	. . . . . 6
16	14, 15	imbi12i 326	. . . . 5
17	16	albii 1640	. . . 4
18	12, 17	bitri 249	. . 3
19	11, 18	bitri 249	. 2
20	10, 19	sylibr 212	1

Syntax hints:  ->wi 4  A.wal 1393  e.wcel 1818  A.wral 2807  C_wss 3475  |^|cint 4286  Trwtr 4545

This theorem is referenced by:  tctr  8192  intwun  9134  intgru  9213  dfon2lem8  29222

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-or 370  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-int 4287  df-tr 4546