Theorem ordunisuc2r 4205

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 2554	. . . . . . . . 9 ⊢ x ∈ V
2	1	sucid 4120	. . . . . . . 8 ⊢ x ∈ suc x
3		elunii 3576	. . . . . . . 8 ⊢ ((x ∈ suc x ∧ suc x ∈ A) → x ∈ ∪ A)
4	2, 3	mpan 400	. . . . . . 7 ⊢ (suc x ∈ A → x ∈ ∪ A)
5	4	imim2i 12	. . . . . 6 ⊢ ((x ∈ A → suc x ∈ A) → (x ∈ A → x ∈ ∪ A))
6	5	alimi 1341	. . . . 5 ⊢ (∀x(x ∈ A → suc x ∈ A) → ∀x(x ∈ A → x ∈ ∪ A))
7		df-ral 2305	. . . . 5 ⊢ (∀x ∈ A suc x ∈ A ↔ ∀x(x ∈ A → suc x ∈ A))
8		dfss2 2928	. . . . 5 ⊢ (A ⊆ ∪ A ↔ ∀x(x ∈ A → x ∈ ∪ A))
9	6, 7, 8	3imtr4i 190	. . . 4 ⊢ (∀x ∈ A suc x ∈ A → A ⊆ ∪ A)
10	9	a1i 9	. . 3 ⊢ (Ord A → (∀x ∈ A suc x ∈ A → A ⊆ ∪ A))
11		orduniss 4128	. . 3 ⊢ (Ord A → ∪ A ⊆ A)
12	10, 11	jctird 300	. 2 ⊢ (Ord A → (∀x ∈ A suc x ∈ A → (A ⊆ ∪ A ∧ ∪ A ⊆ A)))
13		eqss 2954	. 2 ⊢ (A = ∪ A ↔ (A ⊆ ∪ A ∧ ∪ A ⊆ A))
14	12, 13	syl6ibr 151	1 ⊢ (Ord A → (∀x ∈ A suc x ∈ A → A = ∪ A))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911  ∪ cuni 3571  Ord word 4065  suc csuc 4068

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-suc 4074

This theorem is referenced by: (None)