Theorem triun 4237

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 4011	. . . 4 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
2		r19.29 2688	. . . . 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 2392	. . . . . . 7 |- F/_ x y
4		nfiu1 4037	. . . . . . 7 |- F/_ x U_ x e. A B
5	3, 4	nfss 3241	. . . . . 6 |- F/ x y C_ U_ x e. A B
6		trss 4233	. . . . . . . 8 |- ( Tr B -> ( y e. B -> y C_ B ) )
7	6	imp 124	. . . . . . 7 |- ( ( Tr B /\ y e. B ) -> y C_ B )
8		ssiun2 4050	. . . . . . . 8 |- ( x e. A -> B C_ U_ x e. A B )
9		sstr2 3255	. . . . . . . 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 2661	. . . . 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 287	. . 3 |- ( ( A. x e. A Tr B /\ y e. U_ x e. A B ) -> y C_ U_ x e. A B )
15	14	ralrimiva 2623	. 2 |- ( A. x e. A Tr B -> A. y e. U_ x e. A B y C_ U_ x e. A B )
16		dftr3 4228	. 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 134	1 |- ( A. x e. A Tr B -> Tr U_ x e. A B )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   A.wral 2528   E.wrex 2529   C_ wss 3220   U_ciun 4007   Tr wtr 4224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-uni 3931  df-iun 4009  df-tr 4225

This theorem is referenced by:  truni  4238