Theorem triun 3867

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

Assertion

Ref	Expression
triun	|- ( A. x e. A Tr B -> Tr U_ x e. A B )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem triun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 3661	. . . 4 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
2		r19.29 2450	. . . . 5 |- ( ( A. x e. A Tr B /\ E. x e. A y e. B ) -> E. x e. A ( Tr B /\ y e. B ) )
3		nfcv 2178	. . . . . . 7 |- F/_ x y
4		nfiu1 3687	. . . . . . 7 |- F/_ x U_ x e. A B
5	3, 4	nfss 2938	. . . . . 6 |- F/ x y C_ U_ x e. A B
6		trss 3863	. . . . . . . 8 |- ( Tr B -> ( y e. B -> y C_ B ) )
7	6	imp 115	. . . . . . 7 |- ( ( Tr B /\ y e. B ) -> y C_ B )
8		ssiun2 3700	. . . . . . . 8 |- ( x e. A -> B C_ U_ x e. A B )
9		sstr2 2952	. . . . . . . 8 |- ( y C_ B -> ( B C_ U_ x e. A B -> y C_ U_ x e. A B ) )
10	8, 9	syl5com 26	. . . . . . 7 |- ( x e. A -> ( y C_ B -> y C_ U_ x e. A B ) )
11	7, 10	syl5 28	. . . . . 6 |- ( x e. A -> ( ( Tr B /\ y e. B ) -> y C_ U_ x e. A B ) )
12	5, 11	rexlimi 2426	. . . . 5 |- ( E. x e. A ( Tr B /\ y e. B ) -> y C_ U_ x e. A B )
13	2, 12	syl 14	. . . 4 |- ( ( A. x e. A Tr B /\ E. x e. A y e. B ) -> y C_ U_ x e. A B )
14	1, 13	sylan2b 271	. . 3 |- ( ( A. x e. A Tr B /\ y e. U_ x e. A B ) -> y C_ U_ x e. A B )
15	14	ralrimiva 2392	. 2 |- ( A. x e. A Tr B -> A. y e. U_ x e. A B y C_ U_ x e. A B )
16		dftr3 3858	. 2 |- ( Tr U_ x e. A B <-> A. y e. U_ x e. A B y C_ U_ x e. A B )
17	15, 16	sylibr 137	1 |- ( A. x e. A Tr B -> Tr U_ x e. A B )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   A.wral 2306   E.wrex 2307   C_ wss 2917   U_ciun 3657   Tr wtr 3854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-iun 3659  df-tr 3855

This theorem is referenced by:  truni  3868