Theorem bj-ssom 9395

Description: A characterization of subclasses of 𝜔. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ssom	⊢ (∀x(Ind x → A ⊆ x) ↔ A ⊆ 𝜔)

Distinct variable group:   x,A

Proof of Theorem bj-ssom

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 3622	. . 3 ⊢ (A ⊆ ∩ {y ∣ Ind y} ↔ ∀x ∈ {y ∣ Ind y}A ⊆ x)
2		df-ral 2305	. . 3 ⊢ (∀x ∈ {y ∣ Ind y}A ⊆ x ↔ ∀x(x ∈ {y ∣ Ind y} → A ⊆ x))
3		vex 2554	. . . . . 6 ⊢ x ∈ V
4		bj-indeq 9388	. . . . . 6 ⊢ (y = x → (Ind y ↔ Ind x))
5	3, 4	elab 2681	. . . . 5 ⊢ (x ∈ {y ∣ Ind y} ↔ Ind x)
6	5	imbi1i 227	. . . 4 ⊢ ((x ∈ {y ∣ Ind y} → A ⊆ x) ↔ (Ind x → A ⊆ x))
7	6	albii 1356	. . 3 ⊢ (∀x(x ∈ {y ∣ Ind y} → A ⊆ x) ↔ ∀x(Ind x → A ⊆ x))
8	1, 2, 7	3bitrri 196	. 2 ⊢ (∀x(Ind x → A ⊆ x) ↔ A ⊆ ∩ {y ∣ Ind y})
9		bj-dfom 9392	. . . 4 ⊢ 𝜔 = ∩ {y ∣ Ind y}
10	9	eqcomi 2041	. . 3 ⊢ ∩ {y ∣ Ind y} = 𝜔
11	10	sseq2i 2964	. 2 ⊢ (A ⊆ ∩ {y ∣ Ind y} ↔ A ⊆ 𝜔)
12	8, 11	bitri 173	1 ⊢ (∀x(Ind x → A ⊆ x) ↔ A ⊆ 𝜔)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   ∈ wcel 1390  {cab 2023  ∀wral 2300   ⊆ wss 2911  ∩ cint 3606  𝜔com 4256  Ind wind 9385

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-in 2918  df-ss 2925  df-int 3607  df-iom 4257  df-bj-ind 9386

This theorem is referenced by:  bj-om  9396