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Theorem unisucg 6443
Description: A transitive class is equal to the union of its successor, closed form. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.) Generalize from unisuc 6444. (Revised by BJ, 28-Dec-2024.)
Assertion
Ref Expression
unisucg (𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))

Proof of Theorem unisucg
StepHypRef Expression
1 ssequn1 4181 . . 3 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
21a1i 11 . 2 (𝐴𝑉 → ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴))
3 df-tr 5267 . . 3 (Tr 𝐴 𝐴𝐴)
43a1i 11 . 2 (𝐴𝑉 → (Tr 𝐴 𝐴𝐴))
5 unisucs 6442 . . 3 (𝐴𝑉 suc 𝐴 = ( 𝐴𝐴))
65eqeq1d 2735 . 2 (𝐴𝑉 → ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴))
72, 4, 63bitr4d 311 1 (𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  cun 3947  wss 3949   cuni 4909  Tr wtr 5266  suc csuc 6367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-pr 4632  df-uni 4910  df-tr 5267  df-suc 6371
This theorem is referenced by:  unisuc  6444  onunisuc  6475
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