Theorem unisuc 4264

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 3185	. 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
2		df-tr 3959	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
3		df-suc 4222	. . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	unieqi 3685	. . . 4 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
5		uniun 3694	. . . 4 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
6		unisuc.1	. . . . . 6 ⊢ 𝐴 ∈ V
7	6	unisn 3691	. . . . 5 ⊢ ∪ {𝐴} = 𝐴
8	7	uneq2i 3166	. . . 4 ⊢ (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴)
9	4, 5, 8	3eqtri 2119	. . 3 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)
10	9	eqeq1i 2102	. 2 ⊢ (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
11	1, 2, 10	3bitr4i 211	1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)

Syntax hints:   ↔ wb 104   = wceq 1296   ∈ wcel 1445  Vcvv 2633   ∪ cun 3011   ⊆ wss 3013  {csn 3466  ∪ cuni 3675  Tr wtr 3958  suc csuc 4216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-pr 3473  df-uni 3676  df-tr 3959  df-suc 4222

This theorem is referenced by:  onunisuci  4283  ordsucunielexmid  4375  tfrexlem  6137  nnsucuniel  6296