Theorem bj-ssom 15872

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 3901	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥)
2		df-ral 2489	. . 3 ⊢ (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥))
3		vex 2775	. . . . . 6 ⊢ 𝑥 ∈ V
4		bj-indeq 15865	. . . . . 6 ⊢ (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥))
5	3, 4	elab 2917	. . . . 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 1493	. . 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 15869	. . . 4 ⊢ ω = ∩ {𝑦 ∣ Ind 𝑦}
10	9	eqcomi 2209	. . 3 ⊢ ∩ {𝑦 ∣ Ind 𝑦} = ω
11	10	sseq2i 3220	. 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 1371   ∈ wcel 2176  {cab 2191  ∀wral 2484   ⊆ wss 3166  ∩ cint 3885  ωcom 4638  Ind wind 15862

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-int 3886  df-iom 4639  df-bj-ind 15863

This theorem is referenced by:  bj-om  15873