Theorem triun 4100

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 3877	. . . 4 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
2		r19.29 2607	. . . . 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 2312	. . . . . . 7 |- F/_ x y
4		nfiu1 3903	. . . . . . 7 |- F/_ x U_ x e. A B
5	3, 4	nfss 3140	. . . . . 6 |- F/ x y C_ U_ x e. A B
6		trss 4096	. . . . . . . 8 |- ( Tr B -> ( y e. B -> y C_ B ) )
7	6	imp 123	. . . . . . 7 |- ( ( Tr B /\ y e. B ) -> y C_ B )
8		ssiun2 3916	. . . . . . . 8 |- ( x e. A -> B C_ U_ x e. A B )
9		sstr2 3154	. . . . . . . 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 2580	. . . . 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 285	. . 3 |- ( ( A. x e. A Tr B /\ y e. U_ x e. A B ) -> y C_ U_ x e. A B )
15	14	ralrimiva 2543	. 2 |- ( A. x e. A Tr B -> A. y e. U_ x e. A B y C_ U_ x e. A B )
16		dftr3 4091	. 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 133	1 |- ( A. x e. A Tr B -> Tr U_ x e. A B )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   A.wral 2448   E.wrex 2449   C_ wss 3121   U_ciun 3873   Tr wtr 4087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-iun 3875  df-tr 4088

This theorem is referenced by:  truni  4101