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Theorem unisuc 6400
Description: A transitive class is equal to the union of its successor, inference form. 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 unisuc.1 . 2 𝐴 ∈ V
2 unisucg 6399 . 2 (𝐴 ∈ V → (Tr 𝐴 suc 𝐴 = 𝐴))
31, 2ax-mp 5 1 (Tr 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  Vcvv 3447   cuni 4869  Tr wtr 5226  suc csuc 6323
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 3449  df-un 3919  df-in 3921  df-ss 3931  df-sn 4591  df-pr 4593  df-uni 4870  df-tr 5227  df-suc 6327
This theorem is referenced by:  ordunisuc  7771
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