Theorem unisucg 4507

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 4184	. . 3 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		ssequn1 3375	. . 3 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
3	1, 2	bitri 184	. 2 ⊢ (Tr 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
4		df-suc 4464	. . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5	4	unieqi 3899	. . . . 5 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
6		uniun 3908	. . . . 5 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
7	5, 6	eqtri 2250	. . . 4 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ ∪ {𝐴})
8		unisng 3906	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
9	8	uneq2d 3359	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴))
10	7, 9	eqtrid 2274	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))
11	10	eqeq1d 2238	. 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 1395   ∈ wcel 2200   ∪ cun 3196   ⊆ wss 3198  {csn 3667  ∪ cuni 3889  Tr wtr 4183  suc csuc 4458

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-uni 3890  df-tr 4184  df-suc 4464

This theorem is referenced by:  onsucuni2  4658  nlimsucg  4660  ctmlemr  7296  nnnninfeq2  7317  nnsf  16517  peano4nninf  16518  nnnninfex  16534