Theorem unisuc 4330

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 3241	. 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
2		df-tr 4022	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
3		df-suc 4288	. . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	unieqi 3741	. . . 4 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
5		uniun 3750	. . . 4 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
6		unisuc.1	. . . . . 6 ⊢ 𝐴 ∈ V
7	6	unisn 3747	. . . . 5 ⊢ ∪ {𝐴} = 𝐴
8	7	uneq2i 3222	. . . 4 ⊢ (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴)
9	4, 5, 8	3eqtri 2162	. . 3 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)
10	9	eqeq1i 2145	. 2 ⊢ (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
11	1, 2, 10	3bitr4i 211	1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)

Syntax hints:   ↔ wb 104   = wceq 1331   ∈ wcel 1480  Vcvv 2681   ∪ cun 3064   ⊆ wss 3066  {csn 3522  ∪ cuni 3731  Tr wtr 4021  suc csuc 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-uni 3732  df-tr 4022  df-suc 4288

This theorem is referenced by:  onunisuci  4349  ordsucunielexmid  4441  tfrexlem  6224  nnsucuniel  6384