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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 3757 | . . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥) | |
2 | df-ral 2398 | . . 3 ⊢ (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴 ⊆ 𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴 ⊆ 𝑥)) | |
3 | vex 2663 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | bj-indeq 13054 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥)) | |
5 | 3, 4 | elab 2802 | . . . . 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 1431 | . . 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 13058 | . . . 4 ⊢ ω = ∩ {𝑦 ∣ Ind 𝑦} | |
10 | 9 | eqcomi 2121 | . . 3 ⊢ ∩ {𝑦 ∣ Ind 𝑦} = ω |
11 | 10 | sseq2i 3094 | . 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 1314 ∈ wcel 1465 {cab 2103 ∀wral 2393 ⊆ wss 3041 ∩ cint 3741 ωcom 4474 Ind wind 13051 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-in 3047 df-ss 3054 df-int 3742 df-iom 4475 df-bj-ind 13052 |
This theorem is referenced by: bj-om 13062 |
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