Theorem truni 4206

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 4205	. 2 |- ( A. x e. A Tr x -> Tr U_ x e. A x )
2		uniiun 4029	. . 3 |- U. A = U_ x e. A x
3		treq 4198	. . 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 1398   A.wral 2511   U.cuni 3898   U_ciun 3975   Tr wtr 4192

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899  df-iun 3977  df-tr 4193

This theorem is referenced by: (None)