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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-dfom | GIF version |
Description: Alternate definition of ω, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.) |
Ref | Expression |
---|---|
bj-dfom | ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfom3 4333 | . 2 ⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | |
2 | df-bj-ind 10722 | . . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)) | |
3 | 2 | bicomi 130 | . . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) ↔ Ind 𝑥) |
4 | 3 | abbii 2194 | . . 3 ⊢ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = {𝑥 ∣ Ind 𝑥} |
5 | 4 | inteqi 3640 | . 2 ⊢ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = ∩ {𝑥 ∣ Ind 𝑥} |
6 | 1, 5 | eqtri 2101 | 1 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1284 ∈ wcel 1433 {cab 2067 ∀wral 2348 ∅c0 3251 ∩ cint 3636 suc csuc 4120 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-int 3637 df-iom 4332 df-bj-ind 10722 |
This theorem is referenced by: bj-omind 10729 bj-omssind 10730 bj-ssom 10731 |
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