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Theorem unisuc 6020
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 3989 . 2 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
2 df-tr 4954 . 2 (Tr 𝐴 𝐴𝐴)
3 df-suc 5949 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
43unieqi 4646 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5 uniun 4658 . . . 4 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
6 unisuc.1 . . . . . 6 𝐴 ∈ V
76unisn 4653 . . . . 5 {𝐴} = 𝐴
87uneq2i 3970 . . . 4 ( 𝐴 {𝐴}) = ( 𝐴𝐴)
94, 5, 83eqtri 2839 . . 3 suc 𝐴 = ( 𝐴𝐴)
109eqeq1i 2818 . 2 ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴)
111, 2, 103bitr4i 294 1 (Tr 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1637  wcel 2157  Vcvv 3398  cun 3774  wss 3776  {csn 4377   cuni 4637  Tr wtr 4953  suc csuc 5945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rex 3109  df-v 3400  df-un 3781  df-in 3783  df-ss 3790  df-sn 4378  df-pr 4380  df-uni 4638  df-tr 4954  df-suc 5949
This theorem is referenced by:  onunisuci  6057  ordunisuc  7265
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