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Theorem unisuc 6242
 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 4110 . 2 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
2 df-tr 5141 . 2 (Tr 𝐴 𝐴𝐴)
3 df-suc 6172 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
43unieqi 4817 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5 uniun 4827 . . . 4 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
6 unisuc.1 . . . . . 6 𝐴 ∈ V
76unisn 4824 . . . . 5 {𝐴} = 𝐴
87uneq2i 4090 . . . 4 ( 𝐴 {𝐴}) = ( 𝐴𝐴)
94, 5, 83eqtri 2825 . . 3 suc 𝐴 = ( 𝐴𝐴)
109eqeq1i 2803 . 2 ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴)
111, 2, 103bitr4i 306 1 (Tr 𝐴 suc 𝐴 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3442   ∪ cun 3881   ⊆ wss 3883  {csn 4528  ∪ cuni 4804  Tr wtr 5140  suc csuc 6168 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531  df-uni 4805  df-tr 5141  df-suc 6172 This theorem is referenced by:  onunisuci  6280  ordunisuc  7540
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