Theorem unisuc 4415

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 3307	. 2 |- ( U. A C_ A <-> ( U. A u. A ) = A )
2		df-tr 4104	. 2 |- ( Tr A <-> U. A C_ A )
3		df-suc 4373	. . . . 5 |- suc A = ( A u. { A } )
4	3	unieqi 3821	. . . 4 |- U. suc A = U. ( A u. { A } )
5		uniun 3830	. . . 4 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 3827	. . . . 5 |- U. { A } = A
8	7	uneq2i 3288	. . . 4 |- ( U. A u. U. { A } ) = ( U. A u. A )
9	4, 5, 8	3eqtri 2202	. . 3 |- U. suc A = ( U. A u. A )
10	9	eqeq1i 2185	. 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 1353   e. wcel 2148   _Vcvv 2739   u. cun 3129   C_ wss 3131   {csn 3594   U.cuni 3811   Tr wtr 4103   suc csuc 4367

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-suc 4373

This theorem is referenced by:  onunisuci  4434  ordsucunielexmid  4532  tfrexlem  6337  nnsucuniel  6498