Theorem unisuc 4251

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. (Contributed by NM, 30-Aug-1993.)

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 3173	. 2 |- ( U. A C_ A <-> ( U. A u. A ) = A )
2		df-tr 3945	. 2 |- ( Tr A <-> U. A C_ A )
3		df-suc 4209	. . . . 5 |- suc A = ( A u. { A } )
4	3	unieqi 3671	. . . 4 |- U. suc A = U. ( A u. { A } )
5		uniun 3680	. . . 4 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 3677	. . . . 5 |- U. { A } = A
8	7	uneq2i 3154	. . . 4 |- ( U. A u. U. { A } ) = ( U. A u. A )
9	4, 5, 8	3eqtri 2113	. . 3 |- U. suc A = ( U. A u. A )
10	9	eqeq1i 2096	. 2 |- ( U. suc A = A <-> ( U. A u. A ) = A )
11	1, 2, 10	3bitr4i 211	1 |- ( Tr A <-> U. suc A = A )

Syntax hints:   <-> wb 104   = wceq 1290   e. wcel 1439   _Vcvv 2622   u. cun 3000   C_ wss 3002   {csn 3452   U.cuni 3661   Tr wtr 3944   suc csuc 4203

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-sn 3458  df-pr 3459  df-uni 3662  df-tr 3945  df-suc 4209

This theorem is referenced by:  onunisuci  4270  ordsucunielexmid  4362  tfrexlem  6115  nnsucuniel  6272