Theorem unisuc 3046

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 2200	. 2 |- (U.A (_ A <-> (U.A u. A) = A)
2		df-tr 2681	. 2 |- (Tr A <-> U.A (_ A)
3		df-suc 2954	. . . . 5 |- suc A = (A u. {A})
4	3	unieqi 2511	. . . 4 |- U.suc A = U.(A u. {A})
5		uniun 2519	. . . 4 |- U.(A u. {A}) = (U.A u. U.{A})
6		unisuc.1	. . . . . 6 |- A e. V
7	6	unisn 2517	. . . . 5 |- U.{A} = A
8	7	uneq2i 2181	. . . 4 |- (U.A u. U.{A}) = (U.A u. A)
9	4, 5, 8	3eqtr 1499	. . 3 |- U.suc A = (U.A u. A)
10	9	eqeq1i 1482	. 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 956   e. wcel 958  Vcvv 1811   u. cun 2045   (_ wss 2047  {csn 2409  U.cuni 2503  Tr wtr 2680  suc csuc 2950

This theorem is referenced by:  ordunisuc 3089  onunisuc 3106

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-in 2051  df-ss 2053  df-sn 2412  df-pr 2413  df-uni 2504  df-tr 2681  df-suc 2954

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