Theorem unisuc 6391

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 6390	. 2 ⊢ (𝐴 ∈ V → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)

Syntax hints:   ↔ wb 207   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ∪ cuni 4838  Tr wtr 5179  suc csuc 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-un 3888  df-ss 3900  df-sn 4556  df-pr 4558  df-uni 4839  df-tr 5180  df-suc 6316

This theorem is referenced by:  ordunisuc  7772