Theorem unisucg 4505

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 4183	. . 3 |- ( Tr A <-> U. A C_ A )
2		ssequn1 3374	. . 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 4462	. . . . . 6 |- suc A = ( A u. { A } )
5	4	unieqi 3898	. . . . 5 |- U. suc A = U. ( A u. { A } )
6		uniun 3907	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
7	5, 6	eqtri 2250	. . . 4 |- U. suc A = ( U. A u. U. { A } )
8		unisng 3905	. . . . 5 |- ( A e. V -> U. { A } = A )
9	8	uneq2d 3358	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
10	7, 9	eqtrid 2274	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
11	10	eqeq1d 2238	. 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 1395   e. wcel 2200   u. cun 3195   C_ wss 3197   {csn 3666   U.cuni 3888   Tr wtr 4182   suc csuc 4456

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-uni 3889  df-tr 4183  df-suc 4462

This theorem is referenced by:  onsucuni2  4656  nlimsucg  4658  ctmlemr  7275  nnnninfeq2  7296  nnsf  16371  peano4nninf  16372  nnnninfex  16388