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Theorem unisuc 4116
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 3107 . 2 ( AA ↔ ( AA) = A)
2 df-tr 3846 . 2 (Tr A AA)
3 df-suc 4074 . . . . 5 suc A = (A ∪ {A})
43unieqi 3581 . . . 4 suc A = (A ∪ {A})
5 uniun 3590 . . . 4 (A ∪ {A}) = ( A {A})
6 unisuc.1 . . . . . 6 A V
76unisn 3587 . . . . 5 {A} = A
87uneq2i 3088 . . . 4 ( A {A}) = ( AA)
94, 5, 83eqtri 2061 . . 3 suc A = ( AA)
109eqeq1i 2044 . 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 1242   wcel 1390  Vcvv 2551  cun 2909  wss 2911  {csn 3367   cuni 3571  Tr wtr 3845  suc csuc 4068
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-suc 4074
This theorem is referenced by:  onunisuci  4135  ordsucunielexmid  4216  tfrexlem  5889
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