Theorem unisucg 4239

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 4196	. . . . . 6 |- suc A = ( A u. { A } )
2	1	unieqi 3661	. . . . 5 |- U. suc A = U. ( A u. { A } )
3		uniun 3670	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
4	2, 3	eqtri 2108	. . . 4 |- U. suc A = ( U. A u. U. { A } )
5		unisng 3668	. . . . 5 |- ( A e. V -> U. { A } = A )
6	5	uneq2d 3154	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
7	4, 6	syl5eq 2132	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
8	7	eqeq1d 2096	. 2 |- ( A e. V -> ( U. suc A = A <-> ( U. A u. A ) = A ) )
9		df-tr 3935	. . 3 |- ( Tr A <-> U. A C_ A )
10		ssequn1 3170	. . 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 1289   e. wcel 1438   u. cun 2997   C_ wss 2999   {csn 3444   U.cuni 3651   Tr wtr 3934   suc csuc 4190

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3450  df-pr 3451  df-uni 3652  df-tr 3935  df-suc 4196

This theorem is referenced by:  onsucuni2  4378  nlimsucg  4380  nnsf  11778  peano4nninf  11779