Theorem unisucg 4449

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 Jim Kingdon, 18-Aug-2019.)

Assertion

Ref	Expression
unisucg	⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))

Proof of Theorem unisucg

Step	Hyp	Ref	Expression
1		df-tr 4132	. . 3 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		ssequn1 3333	. . 3 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
3	1, 2	bitri 184	. 2 ⊢ (Tr 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
4		df-suc 4406	. . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5	4	unieqi 3849	. . . . 5 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
6		uniun 3858	. . . . 5 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
7	5, 6	eqtri 2217	. . . 4 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ ∪ {𝐴})
8		unisng 3856	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
9	8	uneq2d 3317	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴))
10	7, 9	eqtrid 2241	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))
11	10	eqeq1d 2205	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴))
12	3, 11	bitr4id 199	1 ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2167   ∪ cun 3155   ⊆ wss 3157  {csn 3622  ∪ cuni 3839  Tr wtr 4131  suc csuc 4400

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-uni 3840  df-tr 4132  df-suc 4406

This theorem is referenced by:  onsucuni2  4600  nlimsucg  4602  ctmlemr  7174  nnnninfeq2  7195  nnsf  15649  peano4nninf  15650