Theorem triun 4766

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 4524	. . . 4 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
2		r19.29 3072	. . . . 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 2764	. . . . . . 7 |- F/_ x y
4		nfiu1 4550	. . . . . . 7 |- F/_ x U_ x e. A B
5	3, 4	nfss 3596	. . . . . 6 |- F/ x y C_ U_ x e. A B
6		trss 4761	. . . . . . . 8 |- ( Tr B -> ( y e. B -> y C_ B ) )
7	6	imp 445	. . . . . . 7 |- ( ( Tr B /\ y e. B ) -> y C_ B )
8		ssiun2 4563	. . . . . . 7 |- ( x e. A -> B C_ U_ x e. A B )
9		sstr2 3610	. . . . . . 7 |- ( y C_ B -> ( B C_ U_ x e. A B -> y C_ U_ x e. A B ) )
10	7, 8, 9	syl2imc 41	. . . . . 6 |- ( x e. A -> ( ( Tr B /\ y e. B ) -> y C_ U_ x e. A B ) )
11	5, 10	rexlimi 3024	. . . . 5 |- ( E. x e. A ( Tr B /\ y e. B ) -> y C_ U_ x e. A B )
12	2, 11	syl 17	. . . 4 |- ( ( A. x e. A Tr B /\ E. x e. A y e. B ) -> y C_ U_ x e. A B )
13	1, 12	sylan2b 492	. . 3 |- ( ( A. x e. A Tr B /\ y e. U_ x e. A B ) -> y C_ U_ x e. A B )
14	13	ralrimiva 2966	. 2 |- ( A. x e. A Tr B -> A. y e. U_ x e. A B y C_ U_ x e. A B )
15		dftr3 4756	. 2 |- ( Tr U_ x e. A B <-> A. y e. U_ x e. A B y C_ U_ x e. A B )
16	14, 15	sylibr 224	1 |- ( A. x e. A Tr B -> Tr U_ x e. A B )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   U_ciun 4520   Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-iun 4522  df-tr 4753

This theorem is referenced by:  truni  4767  r1tr  8639  r1elssi  8668  iunord  42422