Theorem unisuc 4444

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 3329	. 2 |- ( U. A C_ A <-> ( U. A u. A ) = A )
2		df-tr 4128	. 2 |- ( Tr A <-> U. A C_ A )
3		df-suc 4402	. . . . 5 |- suc A = ( A u. { A } )
4	3	unieqi 3845	. . . 4 |- U. suc A = U. ( A u. { A } )
5		uniun 3854	. . . 4 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 3851	. . . . 5 |- U. { A } = A
8	7	uneq2i 3310	. . . 4 |- ( U. A u. U. { A } ) = ( U. A u. A )
9	4, 5, 8	3eqtri 2218	. . 3 |- U. suc A = ( U. A u. A )
10	9	eqeq1i 2201	. 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 1364   e. wcel 2164   _Vcvv 2760   u. cun 3151   C_ wss 3153   {csn 3618   U.cuni 3835   Tr wtr 4127   suc csuc 4396

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-suc 4402

This theorem is referenced by:  onunisuci  4463  ordsucunielexmid  4563  tfrexlem  6387  nnsucuniel  6548