Theorem unisucg 4294

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 4251	. . . . . 6 |- suc A = ( A u. { A } )
2	1	unieqi 3710	. . . . 5 |- U. suc A = U. ( A u. { A } )
3		uniun 3719	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
4	2, 3	eqtri 2133	. . . 4 |- U. suc A = ( U. A u. U. { A } )
5		unisng 3717	. . . . 5 |- ( A e. V -> U. { A } = A )
6	5	uneq2d 3194	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
7	4, 6	syl5eq 2157	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
8	7	eqeq1d 2121	. 2 |- ( A e. V -> ( U. suc A = A <-> ( U. A u. A ) = A ) )
9		df-tr 3985	. . 3 |- ( Tr A <-> U. A C_ A )
10		ssequn1 3210	. . 3 |- ( U. A C_ A <-> ( U. A u. A ) = A )
11	9, 10	bitri 183	. 2 |- ( Tr A <-> ( U. A u. A ) = A )
12	8, 11	syl6rbbr 198	1 |- ( A e. V -> ( Tr A <-> U. suc A = A ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1312   e. wcel 1461   u. cun 3033   C_ wss 3035   {csn 3491   U.cuni 3700   Tr wtr 3984   suc csuc 4245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-sn 3497  df-pr 3498  df-uni 3701  df-tr 3985  df-suc 4251

This theorem is referenced by:  onsucuni2  4437  nlimsucg  4439  ctmlemr  6943  nnsf  12880  peano4nninf  12881