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Theorem unisuc 6099
 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 4038 . 2 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
2 df-tr 5025 . 2 (Tr 𝐴 𝐴𝐴)
3 df-suc 6029 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
43unieqi 4715 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5 uniun 4725 . . . 4 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
6 unisuc.1 . . . . . 6 𝐴 ∈ V
76unisn 4722 . . . . 5 {𝐴} = 𝐴
87uneq2i 4019 . . . 4 ( 𝐴 {𝐴}) = ( 𝐴𝐴)
94, 5, 83eqtri 2800 . . 3 suc 𝐴 = ( 𝐴𝐴)
109eqeq1i 2777 . 2 ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴)
111, 2, 103bitr4i 295 1 (Tr 𝐴 suc 𝐴 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1507   ∈ wcel 2050  Vcvv 3409   ∪ cun 3821   ⊆ wss 3823  {csn 4435  ∪ cuni 4706  Tr wtr 5024  suc csuc 6025 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rex 3088  df-v 3411  df-un 3828  df-in 3830  df-ss 3837  df-sn 4436  df-pr 4438  df-uni 4707  df-tr 5025  df-suc 6029 This theorem is referenced by:  onunisuci  6136  ordunisuc  7357
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