Theorem unisuc 4504

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 3374	. 2 |- ( U. A C_ A <-> ( U. A u. A ) = A )
2		df-tr 4183	. 2 |- ( Tr A <-> U. A C_ A )
3		df-suc 4462	. . . . 5 |- suc A = ( A u. { A } )
4	3	unieqi 3898	. . . 4 |- U. suc A = U. ( A u. { A } )
5		uniun 3907	. . . 4 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 3904	. . . . 5 |- U. { A } = A
8	7	uneq2i 3355	. . . 4 |- ( U. A u. U. { A } ) = ( U. A u. A )
9	4, 5, 8	3eqtri 2254	. . 3 |- U. suc A = ( U. A u. A )
10	9	eqeq1i 2237	. 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 1395   e. wcel 2200   _Vcvv 2799   u. cun 3195   C_ wss 3197   {csn 3666   U.cuni 3888   Tr wtr 4182   suc csuc 4456

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-uni 3889  df-tr 4183  df-suc 4462

This theorem is referenced by:  onunisuci  4523  ordsucunielexmid  4623  tfrexlem  6480  nnsucuniel  6641