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Theorem unisuc 5839
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 3816 . 2 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
2 df-tr 4786 . 2 (Tr 𝐴 𝐴𝐴)
3 df-suc 5767 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
43unieqi 4477 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5 uniun 4488 . . . 4 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
6 unisuc.1 . . . . . 6 𝐴 ∈ V
76unisn 4483 . . . . 5 {𝐴} = 𝐴
87uneq2i 3797 . . . 4 ( 𝐴 {𝐴}) = ( 𝐴𝐴)
94, 5, 83eqtri 2677 . . 3 suc 𝐴 = ( 𝐴𝐴)
109eqeq1i 2656 . 2 ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴)
111, 2, 103bitr4i 292 1 (Tr 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  wss 3607  {csn 4210   cuni 4468  Tr wtr 4785  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-pr 4213  df-uni 4469  df-tr 4786  df-suc 5767
This theorem is referenced by:  onunisuci  5879  ordunisuc  7074
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