Theorem unisuc 4391

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 3292	. 2 |- ( U. A C_ A <-> ( U. A u. A ) = A )
2		df-tr 4081	. 2 |- ( Tr A <-> U. A C_ A )
3		df-suc 4349	. . . . 5 |- suc A = ( A u. { A } )
4	3	unieqi 3799	. . . 4 |- U. suc A = U. ( A u. { A } )
5		uniun 3808	. . . 4 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 3805	. . . . 5 |- U. { A } = A
8	7	uneq2i 3273	. . . 4 |- ( U. A u. U. { A } ) = ( U. A u. A )
9	4, 5, 8	3eqtri 2190	. . 3 |- U. suc A = ( U. A u. A )
10	9	eqeq1i 2173	. 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 1343   e. wcel 2136   _Vcvv 2726   u. cun 3114   C_ wss 3116   {csn 3576   U.cuni 3789   Tr wtr 4080   suc csuc 4343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-suc 4349

This theorem is referenced by:  onunisuci  4410  ordsucunielexmid  4508  tfrexlem  6302  nnsucuniel  6463