Theorem truni 4145

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 4144	. 2 |- ( A. x e. A Tr x -> Tr U_ x e. A x )
2		uniiun 3970	. . 3 |- U. A = U_ x e. A x
3		treq 4137	. . 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 1364   A.wral 2475   U.cuni 3839   U_ciun 3916   Tr wtr 4131

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-iun 3918  df-tr 4132

This theorem is referenced by: (None)