Theorem unisuc 4385

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 3287	. 2 |- ( U. A C_ A <-> ( U. A u. A ) = A )
2		df-tr 4075	. 2 |- ( Tr A <-> U. A C_ A )
3		df-suc 4343	. . . . 5 |- suc A = ( A u. { A } )
4	3	unieqi 3793	. . . 4 |- U. suc A = U. ( A u. { A } )
5		uniun 3802	. . . 4 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
6		unisuc.1	. . . . . 6 |- A e. _V
7	6	unisn 3799	. . . . 5 |- U. { A } = A
8	7	uneq2i 3268	. . . 4 |- ( U. A u. U. { A } ) = ( U. A u. A )
9	4, 5, 8	3eqtri 2189	. . 3 |- U. suc A = ( U. A u. A )
10	9	eqeq1i 2172	. 2 |- ( U. suc A = A <-> ( U. A u. A ) = A )
11	1, 2, 10	3bitr4i 211	1 |- ( Tr A <-> U. suc A = A )

Syntax hints:   <-> wb 104   = wceq 1342   e. wcel 2135   _Vcvv 2721   u. cun 3109   C_ wss 3111   {csn 3570   U.cuni 3783   Tr wtr 4074   suc csuc 4337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-sn 3576  df-pr 3577  df-uni 3784  df-tr 4075  df-suc 4343

This theorem is referenced by:  onunisuci  4404  ordsucunielexmid  4502  tfrexlem  6293  nnsucuniel  6454