Theorem unisuc 4168

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 3142	. 2 |- ( U. A C_ A <-> ( U. A u. A ) = A )
2		df-tr 3876	. 2 |- ( Tr A <-> U. A C_ A )
3		df-suc 4126	. . . . 5 |- suc A = ( A u. { A } )
4	3	unieqi 3611	. . . 4 |- U. suc A = U. ( A u. { A } )
5		uniun 3620	. . . 4 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 3617	. . . . 5 |- U. { A } = A
8	7	uneq2i 3123	. . . 4 |- ( U. A u. U. { A } ) = ( U. A u. A )
9	4, 5, 8	3eqtri 2105	. . 3 |- U. suc A = ( U. A u. A )
10	9	eqeq1i 2088	. 2 |- ( U. suc A = A <-> ( U. A u. A ) = A )
11	1, 2, 10	3bitr4i 210	1 |- ( Tr A <-> U. suc A = A )

Syntax hints:   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   u. cun 2971   C_ wss 2973   {csn 3398   U.cuni 3601   Tr wtr 3875   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-suc 4126

This theorem is referenced by:  onunisuci  4187  ordsucunielexmid  4274  tfrexlem  5971