Theorem unisuc 3049

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 2252	. 2 |- (U.A (_ A <-> (U.A u. A) = A)
2		df-tr 2755	. 2 |- (Tr A <-> U.A (_ A)
3		df-suc 2981	. . . . 5 |- suc A = (A u. {A})
4	3	unieqi 2577	. . . 4 |- U.suc A = U.(A u. {A})
5		uniun 2586	. . . 4 |- U.(A u. {A}) = (U.A u. U.{A})
6		unisuc.1	. . . . . 6 |- A e. V
7	6	unisn 2583	. . . . 5 |- U.{A} = A
8	7	uneq2i 2233	. . . 4 |- (U.A u. U.{A}) = (U.A u. A)
9	4, 5, 8	3eqtri 1542	. . 3 |- U.suc A = (U.A u. A)
10	9	eqeq1i 1525	. 2 |- (U.suc A = A <-> (U.A u. A) = A)
11	1, 2, 10	3bitr4i 181	1 |- (Tr A <-> U.suc A = A)

Syntax hints:   <-> wb 144   = wceq 992   e. wcel 994  Vcvv 1857   u. cun 2097   (_ wss 2099  {csn 2467  U.cuni 2569  Tr wtr 2754  suc csuc 2977

This theorem is referenced by:  onunisuci 3083  ordunisuc 3186

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858  df-un 2102  df-in 2103  df-ss 2105  df-sn 2470  df-pr 2471  df-uni 2570  df-tr 2755  df-suc 2981

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