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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-ssom | GIF version |
Description: A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ssom | ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 3652 | . . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥) | |
2 | df-ral 2353 | . . 3 ⊢ (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥)) | |
3 | vex 2604 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | bj-indeq 10724 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥)) | |
5 | 3, 4 | elab 2738 | . . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ Ind 𝑦} ↔ Ind 𝑥) |
6 | 5 | imbi1i 236 | . . . 4 ⊢ ((𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥) ↔ (Ind 𝑥 → 𝐴 ⊆ 𝑥)) |
7 | 6 | albii 1399 | . . 3 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥) ↔ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) |
8 | 1, 2, 7 | 3bitrri 205 | . 2 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦}) |
9 | bj-dfom 10728 | . . . 4 ⊢ ω = ∩ {𝑦 ∣ Ind 𝑦} | |
10 | 9 | eqcomi 2085 | . . 3 ⊢ ∩ {𝑦 ∣ Ind 𝑦} = ω |
11 | 10 | sseq2i 3024 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ 𝐴 ⊆ ω) |
12 | 8, 11 | bitri 182 | 1 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 ∈ wcel 1433 {cab 2067 ∀wral 2348 ⊆ wss 2973 ∩ cint 3636 ωcom 4331 Ind wind 10721 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-in 2979 df-ss 2986 df-int 3637 df-iom 4332 df-bj-ind 10722 |
This theorem is referenced by: bj-om 10732 |
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