Theorem unisuc 3036

Description: A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73.

Hypothesis

Ref	Expression
unisuc.1	|- A e. V

Assertion

Ref	Expression
unisuc	|- (Tr A <-> U.suc A = A)

Proof of Theorem unisuc

Step	Hyp	Ref	Expression
1		ssequn1 2190	. 2 |- (U.A (_ A <-> (U.A u. A) = A)
2		df-tr 2671	. 2 |- (Tr A <-> U.A (_ A)
3		df-suc 2944	. . . . 5 |- suc A = (A u. {A})
4	3	unieqi 2501	. . . 4 |- U.suc A = U.(A u. {A})
5		uniun 2509	. . . 4 |- U.(A u. {A}) = (U.A u. U.{A})
6		unisuc.1	. . . . . 6 |- A e. V
7	6	unisn 2507	. . . . 5 |- U.{A} = A
8	7	uneq2i 2171	. . . 4 |- (U.A u. U.{A}) = (U.A u. A)
9	4, 5, 8	3eqtr 1491	. . 3 |- U.suc A = (U.A u. A)
10	9	eqeq1i 1474	. 2 |- (U.suc A = A <-> (U.A u. A) = A)
11	1, 2, 10	3bitr4 183	1 |- (Tr A <-> U.suc A = A)

Syntax hints:   <-> wb 146   = wceq 953   e. wcel 955  Vcvv 1802   u. cun 2035   (_ wss 2037  {csn 2399  U.cuni 2493  Tr wtr 2670  suc csuc 2940

This theorem is referenced by:  ordunisuc 3079  onunisuc 3096

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-in 2041  df-ss 2043  df-sn 2402  df-pr 2403  df-uni 2494  df-tr 2671  df-suc 2944

Copyright terms: Public domain