Theorem unisucg 4413

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 4101	. . 3 |- ( Tr A <-> U. A C_ A )
2		ssequn1 3305	. . 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 4370	. . . . . 6 |- suc A = ( A u. { A } )
5	4	unieqi 3819	. . . . 5 |- U. suc A = U. ( A u. { A } )
6		uniun 3828	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
7	5, 6	eqtri 2198	. . . 4 |- U. suc A = ( U. A u. U. { A } )
8		unisng 3826	. . . . 5 |- ( A e. V -> U. { A } = A )
9	8	uneq2d 3289	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
10	7, 9	eqtrid 2222	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
11	10	eqeq1d 2186	. 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 1353   e. wcel 2148   u. cun 3127   C_ wss 3129   {csn 3592   U.cuni 3809   Tr wtr 4100   suc csuc 4364

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4101  df-suc 4370

This theorem is referenced by:  onsucuni2  4562  nlimsucg  4564  ctmlemr  7104  nnnninfeq2  7124  nnsf  14614  peano4nninf  14615