Theorem unisuc 4460

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 3343	. 2 |- ( U. A C_ A <-> ( U. A u. A ) = A )
2		df-tr 4143	. 2 |- ( Tr A <-> U. A C_ A )
3		df-suc 4418	. . . . 5 |- suc A = ( A u. { A } )
4	3	unieqi 3860	. . . 4 |- U. suc A = U. ( A u. { A } )
5		uniun 3869	. . . 4 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 3866	. . . . 5 |- U. { A } = A
8	7	uneq2i 3324	. . . 4 |- ( U. A u. U. { A } ) = ( U. A u. A )
9	4, 5, 8	3eqtri 2230	. . 3 |- U. suc A = ( U. A u. A )
10	9	eqeq1i 2213	. 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 1373   e. wcel 2176   _Vcvv 2772   u. cun 3164   C_ wss 3166   {csn 3633   U.cuni 3850   Tr wtr 4142   suc csuc 4412

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4143  df-suc 4418

This theorem is referenced by:  onunisuci  4479  ordsucunielexmid  4579  tfrexlem  6420  nnsucuniel  6581