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Theorem unisuc 4099
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 NM, 30-Aug-1993.)
Hypothesis
Ref Expression
unisuc.1  _V
Assertion
Ref Expression
unisuc  Tr  U. suc

Proof of Theorem unisuc
StepHypRef Expression
1 ssequn1 3090 . 2  U.  C_  U.  u.
2 df-tr 3829 . 2  Tr  U.  C_
3 df-suc 4057 . . . . 5  suc  u.  { }
43unieqi 3564 . . . 4  U. suc  U.  u.  { }
5 uniun 3573 . . . 4  U.  u.  { }  U.  u.  U. { }
6 unisuc.1 . . . . . 6  _V
76unisn 3570 . . . . 5  U. { }
87uneq2i 3071 . . . 4  U.  u.  U. { }  U.  u.
94, 5, 83eqtri 2046 . . 3  U. suc  U.  u.
109eqeq1i 2029 . 2  U. suc  U.  u.
111, 2, 103bitr4i 201 1  Tr  U. suc
Colors of variables: wff set class
Syntax hints:   wb 98   wceq 1228   wcel 1374   _Vcvv 2535    u. cun 2892    C_ wss 2894   {csn 3350   U.cuni 3554   Tr wtr 3828   suc csuc 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-suc 4057
This theorem is referenced by:  onunisuci  4119  ordsucunielexmid  4200  tfrexlem  5870
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