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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 3824 | . . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥) | |
2 | df-ral 2440 | . . 3 ⊢ (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥)) | |
3 | vex 2715 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | bj-indeq 13546 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥)) | |
5 | 3, 4 | elab 2856 | . . . . 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 1450 | . . 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 13550 | . . . 4 ⊢ ω = ∩ {𝑦 ∣ Ind 𝑦} | |
10 | 9 | eqcomi 2161 | . . 3 ⊢ ∩ {𝑦 ∣ Ind 𝑦} = ω |
11 | 10 | sseq2i 3155 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ 𝐴 ⊆ ω) |
12 | 8, 11 | bitri 183 | 1 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1333 ∈ wcel 2128 {cab 2143 ∀wral 2435 ⊆ wss 3102 ∩ cint 3808 ωcom 4550 Ind wind 13543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-v 2714 df-in 3108 df-ss 3115 df-int 3809 df-iom 4551 df-bj-ind 13544 |
This theorem is referenced by: bj-om 13554 |
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