Theorem unisuc 4478

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. (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		ssequn1 3351	. 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
2		df-tr 4159	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
3		df-suc 4436	. . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	unieqi 3874	. . . 4 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
5		uniun 3883	. . . 4 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
6		unisuc.1	. . . . . 6 ⊢ 𝐴 ∈ V
7	6	unisn 3880	. . . . 5 ⊢ ∪ {𝐴} = 𝐴
8	7	uneq2i 3332	. . . 4 ⊢ (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴)
9	4, 5, 8	3eqtri 2232	. . 3 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)
10	9	eqeq1i 2215	. 2 ⊢ (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
11	1, 2, 10	3bitr4i 212	1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1373   ∈ wcel 2178  Vcvv 2776   ∪ cun 3172   ⊆ wss 3174  {csn 3643  ∪ cuni 3864  Tr wtr 4158  suc csuc 4430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-suc 4436

This theorem is referenced by:  onunisuci  4497  ordsucunielexmid  4597  tfrexlem  6443  nnsucuniel  6604