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Theorem unisuc 4122
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 3110 . 2 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
2 df-tr 3852 . 2 (Tr 𝐴 𝐴𝐴)
3 df-suc 4080 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
43unieqi 3587 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5 uniun 3596 . . . 4 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
6 unisuc.1 . . . . . 6 𝐴 ∈ V
76unisn 3593 . . . . 5 {𝐴} = 𝐴
87uneq2i 3091 . . . 4 ( 𝐴 {𝐴}) = ( 𝐴𝐴)
94, 5, 83eqtri 2064 . . 3 suc 𝐴 = ( 𝐴𝐴)
109eqeq1i 2047 . 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 1243  wcel 1393  Vcvv 2554  cun 2912  wss 2914  {csn 3372   cuni 3577  Tr wtr 3851  suc csuc 4074
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-sn 3378  df-pr 3379  df-uni 3578  df-tr 3852  df-suc 4080
This theorem is referenced by:  onunisuci  4141  ordsucunielexmid  4228  tfrexlem  5911
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