Theorem unisucg 4446

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-tr 4129	. . 3 |- ( Tr A <-> U. A C_ A )
2		ssequn1 3330	. . 3 |- ( U. A C_ A <-> ( U. A u. A ) = A )
3	1, 2	bitri 184	. 2 |- ( Tr A <-> ( U. A u. A ) = A )
4		df-suc 4403	. . . . . 6 |- suc A = ( A u. { A } )
5	4	unieqi 3846	. . . . 5 |- U. suc A = U. ( A u. { A } )
6		uniun 3855	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
7	5, 6	eqtri 2214	. . . 4 |- U. suc A = ( U. A u. U. { A } )
8		unisng 3853	. . . . 5 |- ( A e. V -> U. { A } = A )
9	8	uneq2d 3314	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
10	7, 9	eqtrid 2238	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
11	10	eqeq1d 2202	. 2 |- ( A e. V -> ( U. suc A = A <-> ( U. A u. A ) = A ) )
12	3, 11	bitr4id 199	1 |- ( A e. V -> ( Tr A <-> U. suc A = A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   u. cun 3152   C_ wss 3154   {csn 3619   U.cuni 3836   Tr wtr 4128   suc csuc 4397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-uni 3837  df-tr 4129  df-suc 4403

This theorem is referenced by:  onsucuni2  4597  nlimsucg  4599  ctmlemr  7169  nnnninfeq2  7190  nnsf  15565  peano4nninf  15566