Theorem triun 4558

Description: The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
triun

Distinct variable group:   ,

Proof of Theorem triun

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4335	. . . 4
2		r19.29 2992	. . . . 5
3		nfcv 2619	. . . . . . 7
4		nfiu1 4360	. . . . . . 7
5	3, 4	nfss 3496	. . . . . 6
6		trss 4554	. . . . . . . 8
7	6	imp 429	. . . . . . 7
8		ssiun2 4373	. . . . . . . 8
9		sstr2 3510	. . . . . . . 8
10	8, 9	syl5com 30	. . . . . . 7
11	7, 10	syl5 32	. . . . . 6
12	5, 11	rexlimi 2939	. . . . 5
13	2, 12	syl 16	. . . 4
14	1, 13	sylan2b 475	. . 3
15	14	ralrimiva 2871	. 2
16		dftr3 4549	. 2
17	15, 16	sylibr 212	1

Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  A.wral 2807  E.wrex 2808  C_wss 3475  U_ciun 4330  Trwtr 4545

This theorem is referenced by:  truni  4559  r1tr  8215  r1elssi  8244

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-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-uni 4250  df-iun 4332  df-tr 4546