Theorem unisucg 4151

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-suc 4108	. . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	unieqi 3590	. . . . 5 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
3		uniun 3599	. . . . 5 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
4	2, 3	eqtri 2060	. . . 4 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ ∪ {𝐴})
5		unisng 3597	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
6	5	uneq2d 3097	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴))
7	4, 6	syl5eq 2084	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))
8	7	eqeq1d 2048	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴))
9		df-tr 3855	. . 3 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
10		ssequn1 3113	. . 3 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
11	9, 10	bitri 173	. 2 ⊢ (Tr 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
12	8, 11	syl6rbbr 188	1 ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393   ∪ cun 2915   ⊆ wss 2917  {csn 3375  ∪ cuni 3580  Tr wtr 3854  suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-suc 4108

This theorem is referenced by:  onsucuni2  4288  nlimsucg  4290