Theorem unisuc 5801

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 3783	. 2 |- ( U. A C_ A <-> ( U. A u. A ) = A )
2		df-tr 4753	. 2 |- ( Tr A <-> U. A C_ A )
3		df-suc 5729	. . . . 5 |- suc A = ( A u. { A } )
4	3	unieqi 4445	. . . 4 |- U. suc A = U. ( A u. { A } )
5		uniun 4456	. . . 4 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 4451	. . . . 5 |- U. { A } = A
8	7	uneq2i 3764	. . . 4 |- ( U. A u. U. { A } ) = ( U. A u. A )
9	4, 5, 8	3eqtri 2648	. . 3 |- U. suc A = ( U. A u. A )
10	9	eqeq1i 2627	. 2 |- ( U. suc A = A <-> ( U. A u. A ) = A )
11	1, 2, 10	3bitr4i 292	1 |- ( Tr A <-> U. suc A = A )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   U.cuni 4436   Tr wtr 4752   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180  df-uni 4437  df-tr 4753  df-suc 5729

This theorem is referenced by:  onunisuci  5841  ordunisuc  7032