Theorem truni 4155

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 4154	. 2 |- ( A. x e. A Tr x -> Tr U_ x e. A x )
2		uniiun 3980	. . 3 |- U. A = U_ x e. A x
3		treq 4147	. . 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 134	1 |- ( A. x e. A Tr x -> Tr U. A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   A.wral 2483   U.cuni 3849   U_ciun 3926   Tr wtr 4141

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-in 3171  df-ss 3178  df-uni 3850  df-iun 3928  df-tr 4142

This theorem is referenced by: (None)