Theorem unisuc 6462

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 6461	. 2 ⊢ (𝐴 ∈ V → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1539   ∈ wcel 2107  Vcvv 3479  ∪ cuni 4906  Tr wtr 5258  suc csuc 6385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-ss 3967  df-sn 4626  df-pr 4628  df-uni 4907  df-tr 5259  df-suc 6389

This theorem is referenced by:  ordunisuc  7853