Theorem bj-ssom 14691

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 3861	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥)
2		df-ral 2460	. . 3 ⊢ (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥))
3		vex 2741	. . . . . 6 ⊢ 𝑥 ∈ V
4		bj-indeq 14684	. . . . . 6 ⊢ (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥))
5	3, 4	elab 2882	. . . . 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 1470	. . 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 14688	. . . 4 ⊢ ω = ∩ {𝑦 ∣ Ind 𝑦}
10	9	eqcomi 2181	. . 3 ⊢ ∩ {𝑦 ∣ Ind 𝑦} = ω
11	10	sseq2i 3183	. 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 1351   ∈ wcel 2148  {cab 2163  ∀wral 2455   ⊆ wss 3130  ∩ cint 3845  ωcom 4590  Ind wind 14681

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2740  df-in 3136  df-ss 3143  df-int 3846  df-iom 4591  df-bj-ind 14682

This theorem is referenced by:  bj-om  14692