Theorem unisuc 2299

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 1628	. 2 |- (U.A (_ A <-> (U.A u. A) = A)
2		df-tr 2042	. 2 |- (Tr A <-> U.A (_ A)
3		df-suc 2205	. . . . 5 |- suc A = (A u. {A})
4	3	unieqi 1928	. . . 4 |- U.suc A = U.(A u. {A})
5		uniun 1934	. . . 4 |- U.(A u. {A}) = (U.A u. U.{A})
6		unisuc.1	. . . . . 6 |- A e. V
7	6	unisn 1932	. . . . 5 |- U.{A} = A
8	7	uneq2i 1608	. . . 4 |- (U.A u. U.{A}) = (U.A u. A)
9	4, 5, 8	3eqtr 1123	. . 3 |- U.suc A = (U.A u. A)
10	9	cleq1i 1108	. 2 |- (U.suc A = A <-> (U.A u. A) = A)
11	1, 2, 10	3bitr4 158	1 |- (Tr A <-> U.suc A = A)

Syntax hints:   <-> wb 127   = wceq 1091   e. wcel 1092  Vcvv 1348   u. cun 1485   (_ wss 1487  {csn 1808  U.cuni 1919  Tr wtr 2041  suc csuc 2201

This theorem is referenced by:  ordunisuc 2339  onunisuc 2354  nlimsuc 2363

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-sn 1811  df-pr 1812  df-uni 1920  df-tr 2042  df-suc 2205

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