Theorem unisuc 6443

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

Step	Hyp	Ref	Expression
1		unisuc.1	. 2 ⊢ 𝐴 ∈ V
2		unisucg 6442	. 2 ⊢ (𝐴 ∈ V → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)

Syntax hints:   ↔ wb 209   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∪ cuni 4876  Tr wtr 5222  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-sn 4595  df-pr 4597  df-uni 4877  df-tr 5223  df-suc 6367

This theorem is referenced by:  ordunisuc  7827