Theorem triun 3888

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 3682	. . . 4 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
2		r19.29 2494	. . . . 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 2219	. . . . . . 7 |- F/_ x y
4		nfiu1 3708	. . . . . . 7 |- F/_ x U_ x e. A B
5	3, 4	nfss 2992	. . . . . 6 |- F/ x y C_ U_ x e. A B
6		trss 3884	. . . . . . . 8 |- ( Tr B -> ( y e. B -> y C_ B ) )
7	6	imp 122	. . . . . . 7 |- ( ( Tr B /\ y e. B ) -> y C_ B )
8		ssiun2 3721	. . . . . . . 8 |- ( x e. A -> B C_ U_ x e. A B )
9		sstr2 3006	. . . . . . . 8 |- ( y C_ B -> ( B C_ U_ x e. A B -> y C_ U_ x e. A B ) )
10	8, 9	syl5com 29	. . . . . . 7 |- ( x e. A -> ( y C_ B -> y C_ U_ x e. A B ) )
11	7, 10	syl5 32	. . . . . 6 |- ( x e. A -> ( ( Tr B /\ y e. B ) -> y C_ U_ x e. A B ) )
12	5, 11	rexlimi 2470	. . . . 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 281	. . 3 |- ( ( A. x e. A Tr B /\ y e. U_ x e. A B ) -> y C_ U_ x e. A B )
15	14	ralrimiva 2434	. 2 |- ( A. x e. A Tr B -> A. y e. U_ x e. A B y C_ U_ x e. A B )
16		dftr3 3879	. 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 132	1 |- ( A. x e. A Tr B -> Tr U_ x e. A B )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   A.wral 2348   E.wrex 2349   C_ wss 2973   U_ciun 3678   Tr wtr 3875

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-iun 3680  df-tr 3876

This theorem is referenced by:  truni  3889