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Theorem unisuc 4073
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 A V
Assertion
Ref Expression
unisuc (Tr A suc A = A)

Proof of Theorem unisuc
StepHypRef Expression
1 ssequn1 3091 . 2 ( AA ↔ ( AA) = A)
2 df-tr 3807 . 2 (Tr A AA)
3 df-suc 4031 . . . . 5 suc A = (A ∪ {A})
43unieqi 3542 . . . 4 suc A = (A ∪ {A})
5 uniun 3551 . . . 4 (A ∪ {A}) = ( A {A})
6 unisuc.1 . . . . . 6 A V
76unisn 3548 . . . . 5 {A} = A
87uneq2i 3072 . . . 4 ( A {A}) = ( AA)
94, 5, 83eqtri 2046 . . 3 suc A = ( AA)
109eqeq1i 2029 . 2 ( suc A = A ↔ ( AA) = A)
111, 2, 103bitr4i 201 1 (Tr A suc A = A)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1373   wcel 1375  Vcvv 2533  cun 2893  wss 2895  {csn 3327   cuni 3532  Tr wtr 3806  suc csuc 4026
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2288  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-sn 3333  df-pr 3334  df-uni 3533  df-tr 3807  df-suc 4031
This theorem is referenced by:  onunisuci  4092  ordsucunielexmid  4173  tfrexlem  5835
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