Theorem unisuc 4553

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 3399	. 2 |- ( U. A C_ A <-> ( U. A u. A ) = A )
2		df-tr 4225	. 2 |- ( Tr A <-> U. A C_ A )
3		df-suc 4511	. . . . 5 |- suc A = ( A u. { A } )
4	3	unieqi 3940	. . . 4 |- U. suc A = U. ( A u. { A } )
5		uniun 3949	. . . 4 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 3946	. . . . 5 |- U. { A } = A
8	7	uneq2i 3380	. . . 4 |- ( U. A u. U. { A } ) = ( U. A u. A )
9	4, 5, 8	3eqtri 2263	. . 3 |- U. suc A = ( U. A u. A )
10	9	eqeq1i 2246	. 2 |- ( U. suc A = A <-> ( U. A u. A ) = A )
11	1, 2, 10	3bitr4i 212	1 |- ( Tr A <-> U. suc A = A )

Syntax hints:   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   C_ wss 3220   {csn 3705   U.cuni 3930   Tr wtr 4224   suc csuc 4505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-suc 4511

This theorem is referenced by:  onunisuci  4572  ordsucunielexmid  4673  tfrexlem  6595  nnsucuniel  6758