Theorem unisuc 2299

Description: A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73.

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		ssequn1 1628	. 2 ⊢ (∪A ⊆ A ↔ (∪A ∪ A) = A)
2		df-tr 2042	. 2 ⊢ (Tr A ↔ ∪A ⊆ A)
3		df-suc 2205	. . . . 5 ⊢ suc A = (A ∪ {A})
4	3	unieqi 1928	. . . 4 ⊢ ∪suc A = ∪(A ∪ {A})
5		uniun 1934	. . . 4 ⊢ ∪(A ∪ {A}) = (∪A ∪ ∪{A})
6		unisuc.1	. . . . . 6 ⊢ A ∈ V
7	6	unisn 1932	. . . . 5 ⊢ ∪{A} = A
8	7	uneq2i 1608	. . . 4 ⊢ (∪A ∪ ∪{A}) = (∪A ∪ A)
9	4, 5, 8	3eqtr 1123	. . 3 ⊢ ∪suc A = (∪A ∪ A)
10	9	cleq1i 1108	. 2 ⊢ (∪suc A = A ↔ (∪A ∪ A) = A)
11	1, 2, 10	3bitr4 158	1 ⊢ (Tr A ↔ ∪suc A = A)

Syntax hints:   ↔ wb 127   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ⊆ wss 1487  {csn 1808  ∪cuni 1919  Tr wtr 2041  suc csuc 2201

This theorem is referenced by:  ordunisuc 2339  onunisuc 2354  nlimsuc 2363

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-sn 1811  df-pr 1812  df-uni 1920  df-tr 2042  df-suc 2205

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