Theorem unisuc 6427

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 6426	. 2 ⊢ (𝐴 ∈ V → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)

Syntax hints:   ↔ wb 208   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ∪ cuni 4865  Tr wtr 5207  suc csuc 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-ss 3921  df-sn 4583  df-pr 4585  df-uni 4866  df-tr 5208  df-suc 6352

This theorem is referenced by:  ordunisuc  7812