Theorem unisucg 4479

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 4159	. . 3 |- ( Tr A <-> U. A C_ A )
2		ssequn1 3351	. . 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 4436	. . . . . 6 |- suc A = ( A u. { A } )
5	4	unieqi 3874	. . . . 5 |- U. suc A = U. ( A u. { A } )
6		uniun 3883	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
7	5, 6	eqtri 2228	. . . 4 |- U. suc A = ( U. A u. U. { A } )
8		unisng 3881	. . . . 5 |- ( A e. V -> U. { A } = A )
9	8	uneq2d 3335	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
10	7, 9	eqtrid 2252	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
11	10	eqeq1d 2216	. 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 1373   e. wcel 2178   u. cun 3172   C_ wss 3174   {csn 3643   U.cuni 3864   Tr wtr 4158   suc csuc 4430

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-suc 4436

This theorem is referenced by:  onsucuni2  4630  nlimsucg  4632  ctmlemr  7236  nnnninfeq2  7257  nnsf  16144  peano4nninf  16145  nnnninfex  16161