Theorem unisucg 4169

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 Jim Kingdon, 18-Aug-2019.)

Assertion

Ref	Expression
unisucg	|- ( A e. V -> ( Tr A <-> U. suc A = A ) )

Proof of Theorem unisucg

Step	Hyp	Ref	Expression
1		df-suc 4126	. . . . . 6 |- suc A = ( A u. { A } )
2	1	unieqi 3611	. . . . 5 |- U. suc A = U. ( A u. { A } )
3		uniun 3620	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
4	2, 3	eqtri 2101	. . . 4 |- U. suc A = ( U. A u. U. { A } )
5		unisng 3618	. . . . 5 |- ( A e. V -> U. { A } = A )
6	5	uneq2d 3126	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
7	4, 6	syl5eq 2125	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
8	7	eqeq1d 2089	. 2 |- ( A e. V -> ( U. suc A = A <-> ( U. A u. A ) = A ) )
9		df-tr 3876	. . 3 |- ( Tr A <-> U. A C_ A )
10		ssequn1 3142	. . 3 |- ( U. A C_ A <-> ( U. A u. A ) = A )
11	9, 10	bitri 182	. 2 |- ( Tr A <-> ( U. A u. A ) = A )
12	8, 11	syl6rbbr 197	1 |- ( A e. V -> ( Tr A <-> U. suc A = A ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   u. cun 2971   C_ wss 2973   {csn 3398   U.cuni 3601   Tr wtr 3875   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-suc 4126

This theorem is referenced by:  onsucuni2  4307  nlimsucg  4309