Theorem ordunisuc2r 4438

Description: An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.)

Assertion

Ref	Expression
ordunisuc2r	⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 = ∪ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem ordunisuc2r

Step	Hyp	Ref	Expression
1		vex 2692	. . . . . . . . 9 ⊢ 𝑥 ∈ V
2	1	sucid 4347	. . . . . . . 8 ⊢ 𝑥 ∈ suc 𝑥
3		elunii 3749	. . . . . . . 8 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐴)
4	2, 3	mpan 421	. . . . . . 7 ⊢ (suc 𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)
5	4	imim2i 12	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴))
6	5	alimi 1432	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴))
7		df-ral 2422	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
8		dfss2 3091	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴))
9	6, 7, 8	3imtr4i 200	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴)
10	9	a1i 9	. . 3 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴))
11		orduniss 4355	. . 3 ⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴)
12	10, 11	jctird 315	. 2 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → (𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝐴)))
13		eqss 3117	. 2 ⊢ (𝐴 = ∪ 𝐴 ↔ (𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝐴))
14	12, 13	syl6ibr 161	1 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 = ∪ 𝐴))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332   ∈ wcel 1481  ∀wral 2417   ⊆ wss 3076  ∪ cuni 3744  Ord word 4292  suc csuc 4295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-uni 3745  df-tr 4035  df-iord 4296  df-suc 4301

This theorem is referenced by: (None)