Theorem truni 4101

Description: The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)

Assertion

Ref	Expression
truni	|- ( A. x e. A Tr x -> Tr U. A )

Distinct variable group:   x, A

Proof of Theorem truni

Step	Hyp	Ref	Expression
1		triun 4100	. 2 |- ( A. x e. A Tr x -> Tr U_ x e. A x )
2		uniiun 3926	. . 3 |- U. A = U_ x e. A x
3		treq 4093	. . 3 |- ( U. A = U_ x e. A x -> ( Tr U. A <-> Tr U_ x e. A x ) )
4	2, 3	ax-mp 5	. 2 |- ( Tr U. A <-> Tr U_ x e. A x )
5	1, 4	sylibr 133	1 |- ( A. x e. A Tr x -> Tr U. A )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   A.wral 2448   U.cuni 3796   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: (None)