Theorem unisucg 4177

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 4134	. . . . . 6 |- suc A = ( A u. { A } )
2	1	unieqi 3619	. . . . 5 |- U. suc A = U. ( A u. { A } )
3		uniun 3628	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
4	2, 3	eqtri 2102	. . . 4 |- U. suc A = ( U. A u. U. { A } )
5		unisng 3626	. . . . 5 |- ( A e. V -> U. { A } = A )
6	5	uneq2d 3127	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
7	4, 6	syl5eq 2126	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
8	7	eqeq1d 2090	. 2 |- ( A e. V -> ( U. suc A = A <-> ( U. A u. A ) = A ) )
9		df-tr 3884	. . 3 |- ( Tr A <-> U. A C_ A )
10		ssequn1 3143	. . 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 1285   e. wcel 1434   u. cun 2972   C_ wss 2974   {csn 3406   U.cuni 3609   Tr wtr 3883   suc csuc 4128

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-uni 3610  df-tr 3884  df-suc 4134

This theorem is referenced by:  onsucuni2  4315  nlimsucg  4317