Theorem bj-ssom 15428

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 3886	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥)
2		df-ral 2477	. . 3 ⊢ (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥))
3		vex 2763	. . . . . 6 ⊢ 𝑥 ∈ V
4		bj-indeq 15421	. . . . . 6 ⊢ (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥))
5	3, 4	elab 2904	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ Ind 𝑦} ↔ Ind 𝑥)
6	5	imbi1i 238	. . . 4 ⊢ ((𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥) ↔ (Ind 𝑥 → 𝐴 ⊆ 𝑥))
7	6	albii 1481	. . 3 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥) ↔ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥))
8	1, 2, 7	3bitrri 207	. 2 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦})
9		bj-dfom 15425	. . . 4 ⊢ ω = ∩ {𝑦 ∣ Ind 𝑦}
10	9	eqcomi 2197	. . 3 ⊢ ∩ {𝑦 ∣ Ind 𝑦} = ω
11	10	sseq2i 3206	. 2 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ 𝐴 ⊆ ω)
12	8, 11	bitri 184	1 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   ∈ wcel 2164  {cab 2179  ∀wral 2472   ⊆ wss 3153  ∩ cint 3870  ωcom 4622  Ind wind 15418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-in 3159  df-ss 3166  df-int 3871  df-iom 4623  df-bj-ind 15419

This theorem is referenced by:  bj-om  15429