Theorem unisucg 4151

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 4108	. . . . . 6 |- suc A = ( A u. { A } )
2	1	unieqi 3590	. . . . 5 |- U. suc A = U. ( A u. { A } )
3		uniun 3599	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
4	2, 3	eqtri 2060	. . . 4 |- U. suc A = ( U. A u. U. { A } )
5		unisng 3597	. . . . 5 |- ( A e. V -> U. { A } = A )
6	5	uneq2d 3097	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
7	4, 6	syl5eq 2084	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
8	7	eqeq1d 2048	. 2 |- ( A e. V -> ( U. suc A = A <-> ( U. A u. A ) = A ) )
9		df-tr 3855	. . 3 |- ( Tr A <-> U. A C_ A )
10		ssequn1 3113	. . 3 |- ( U. A C_ A <-> ( U. A u. A ) = A )
11	9, 10	bitri 173	. 2 |- ( Tr A <-> ( U. A u. A ) = A )
12	8, 11	syl6rbbr 188	1 |- ( A e. V -> ( Tr A <-> U. suc A = A ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   u. cun 2915   C_ wss 2917   {csn 3375   U.cuni 3580   Tr wtr 3854   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-suc 4108

This theorem is referenced by:  onsucuni2  4288  nlimsucg  4290