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Theorem unisuc 6394
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 6393 . 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 1541  wcel 2106  Vcvv 3443   cuni 4863  Tr wtr 5220  suc csuc 6317
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3445  df-un 3913  df-in 3915  df-ss 3925  df-sn 4585  df-pr 4587  df-uni 4864  df-tr 5221  df-suc 6321
This theorem is referenced by:  ordunisuc  7759
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