Theorem unisucg 4399

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 4088	. . 3 |- ( Tr A <-> U. A C_ A )
2		ssequn1 3297	. . 3 |- ( U. A C_ A <-> ( U. A u. A ) = A )
3	1, 2	bitri 183	. 2 |- ( Tr A <-> ( U. A u. A ) = A )
4		df-suc 4356	. . . . . 6 |- suc A = ( A u. { A } )
5	4	unieqi 3806	. . . . 5 |- U. suc A = U. ( A u. { A } )
6		uniun 3815	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
7	5, 6	eqtri 2191	. . . 4 |- U. suc A = ( U. A u. U. { A } )
8		unisng 3813	. . . . 5 |- ( A e. V -> U. { A } = A )
9	8	uneq2d 3281	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
10	7, 9	eqtrid 2215	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
11	10	eqeq1d 2179	. 2 |- ( A e. V -> ( U. suc A = A <-> ( U. A u. A ) = A ) )
12	3, 11	bitr4id 198	1 |- ( A e. V -> ( Tr A <-> U. suc A = A ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   u. cun 3119   C_ wss 3121   {csn 3583   U.cuni 3796   Tr wtr 4087   suc csuc 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-suc 4356

This theorem is referenced by:  onsucuni2  4548  nlimsucg  4550  ctmlemr  7085  nnnninfeq2  7105  nnsf  14038  peano4nninf  14039