Theorem unisuc 3052

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 2203	. 2 ⊢ (∪A ⊆ A ↔ (∪A ∪ A) = A)
2		df-tr 2686	. 2 ⊢ (Tr A ↔ ∪A ⊆ A)
3		df-suc 2960	. . . . 5 ⊢ suc A = (A ∪ {A})
4	3	unieqi 2515	. . . 4 ⊢ ∪suc A = ∪(A ∪ {A})
5		uniun 2523	. . . 4 ⊢ ∪(A ∪ {A}) = (∪A ∪ ∪{A})
6		unisuc.1	. . . . . 6 ⊢ A ∈ V
7	6	unisn 2521	. . . . 5 ⊢ ∪{A} = A
8	7	uneq2i 2184	. . . 4 ⊢ (∪A ∪ ∪{A}) = (∪A ∪ A)
9	4, 5, 8	3eqtr 1502	. . 3 ⊢ ∪suc A = (∪A ∪ A)
10	9	eqeq1i 1485	. 2 ⊢ (∪suc A = A ↔ (∪A ∪ A) = A)
11	1, 2, 10	3bitr4 183	1 ⊢ (Tr A ↔ ∪suc A = A)

Syntax hints:   ↔ wb 146   = wceq 958   ∈ wcel 960  Vcvv 1814   ∪ cun 2048   ⊆ wss 2050  {csn 2413  ∪cuni 2507  Tr wtr 2685  suc csuc 2956

This theorem is referenced by:  ordunisuc 3095  onunisuc 3112

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054  df-ss 2056  df-sn 2416  df-pr 2417  df-uni 2508  df-tr 2686  df-suc 2960

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