Theorem ordunisuc2r 4260

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 2605	. . . . . . . . 9 ⊢ 𝑥 ∈ V
2	1	sucid 4174	. . . . . . . 8 ⊢ 𝑥 ∈ suc 𝑥
3		elunii 3608	. . . . . . . 8 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐴)
4	2, 3	mpan 415	. . . . . . 7 ⊢ (suc 𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)
5	4	imim2i 12	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴))
6	5	alimi 1385	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴))
7		df-ral 2354	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
8		dfss2 2989	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴))
9	6, 7, 8	3imtr4i 199	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴)
10	9	a1i 9	. . 3 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴))
11		orduniss 4182	. . 3 ⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴)
12	10, 11	jctird 310	. 2 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → (𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝐴)))
13		eqss 3015	. 2 ⊢ (𝐴 = ∪ 𝐴 ↔ (𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝐴))
14	12, 13	syl6ibr 160	1 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 = ∪ 𝐴))

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1283   = wceq 1285   ∈ wcel 1434  ∀wral 2349   ⊆ wss 2974  ∪ cuni 3603  Ord word 4119  suc csuc 4122

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-uni 3604  df-tr 3878  df-iord 4123  df-suc 4128

This theorem is referenced by: (None)