Theorem unisuc 4516

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 3379	. 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
2		df-tr 4193	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
3		df-suc 4474	. . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	unieqi 3908	. . . 4 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
5		uniun 3917	. . . 4 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
6		unisuc.1	. . . . . 6 ⊢ 𝐴 ∈ V
7	6	unisn 3914	. . . . 5 ⊢ ∪ {𝐴} = 𝐴
8	7	uneq2i 3360	. . . 4 ⊢ (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴)
9	4, 5, 8	3eqtri 2256	. . 3 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)
10	9	eqeq1i 2239	. 2 ⊢ (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
11	1, 2, 10	3bitr4i 212	1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ∪ cun 3199   ⊆ wss 3201  {csn 3673  ∪ cuni 3898  Tr wtr 4192  suc csuc 4468

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-uni 3899  df-tr 4193  df-suc 4474

This theorem is referenced by:  onunisuci  4535  ordsucunielexmid  4635  tfrexlem  6543  nnsucuniel  6706