Theorem unisuc 3741

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 2775	. 2 |- (U.A C_ A <-> (U.A u. A) = A)
2		df-tr 3412	. 2 |- (Tr A <-> U.A C_ A)
3		df-suc 3663	. . . . 5 |- suc A = (A u. {A})
4	3	unieqi 3187	. . . 4 |- U.suc A = U.(A u. {A})
5		uniun 3196	. . . 4 |- U.(A u. {A}) = (U.A u. U.{A})
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 3193	. . . . 5 |- U.{A} = A
8	7	uneq2i 2752	. . . 4 |- (U.A u. U.{A}) = (U.A u. A)
9	4, 5, 8	3eqtri 1912	. . 3 |- U.suc A = (U.A u. A)
10	9	eqeq1i 1891	. 2 |- (U.suc A = A <-> (U.A u. A) = A)
11	1, 2, 10	3bitr4i 200	1 |- (Tr A <-> U.suc A = A)

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  _Vcvv 2292   u. cun 2591   C_ wss 2593  {csn 3044  U.cuni 3177  Tr wtr 3411  suc csuc 3659

This theorem is referenced by:  onunisuci 3783  ordunisuc 3911  ordunisucOLD 3912

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294  df-un 2600  df-in 2603  df-ss 2605  df-sn 3049  df-pr 3050  df-uni 3178  df-tr 3412  df-suc 3663

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