Theorem bj-ssom 12826

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 3753	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥)
2		df-ral 2395	. . 3 ⊢ (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥))
3		vex 2660	. . . . . 6 ⊢ 𝑥 ∈ V
4		bj-indeq 12819	. . . . . 6 ⊢ (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥))
5	3, 4	elab 2798	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ Ind 𝑦} ↔ Ind 𝑥)
6	5	imbi1i 237	. . . 4 ⊢ ((𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥) ↔ (Ind 𝑥 → 𝐴 ⊆ 𝑥))
7	6	albii 1429	. . 3 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥) ↔ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥))
8	1, 2, 7	3bitrri 206	. 2 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦})
9		bj-dfom 12823	. . . 4 ⊢ ω = ∩ {𝑦 ∣ Ind 𝑦}
10	9	eqcomi 2119	. . 3 ⊢ ∩ {𝑦 ∣ Ind 𝑦} = ω
11	10	sseq2i 3090	. 2 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ 𝐴 ⊆ ω)
12	8, 11	bitri 183	1 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1312   ∈ wcel 1463  {cab 2101  ∀wral 2390   ⊆ wss 3037  ∩ cint 3737  ωcom 4464  Ind wind 12816

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-in 3043  df-ss 3050  df-int 3738  df-iom 4465  df-bj-ind 12817

This theorem is referenced by:  bj-om  12827