Theorem unisucg 4450

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 4133	. . 3 |- ( Tr A <-> U. A C_ A )
2		ssequn1 3334	. . 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 4407	. . . . . 6 |- suc A = ( A u. { A } )
5	4	unieqi 3850	. . . . 5 |- U. suc A = U. ( A u. { A } )
6		uniun 3859	. . . . 5 |- U. ( A u. { A } ) = ( U. A u. U. { A } )
7	5, 6	eqtri 2217	. . . 4 |- U. suc A = ( U. A u. U. { A } )
8		unisng 3857	. . . . 5 |- ( A e. V -> U. { A } = A )
9	8	uneq2d 3318	. . . 4 |- ( A e. V -> ( U. A u. U. { A } ) = ( U. A u. A ) )
10	7, 9	eqtrid 2241	. . 3 |- ( A e. V -> U. suc A = ( U. A u. A ) )
11	10	eqeq1d 2205	. 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 2167   u. cun 3155   C_ wss 3157   {csn 3623   U.cuni 3840   Tr wtr 4132   suc csuc 4401

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-uni 3841  df-tr 4133  df-suc 4407

This theorem is referenced by:  onsucuni2  4601  nlimsucg  4603  ctmlemr  7183  nnnninfeq2  7204  nnsf  15736  peano4nninf  15737