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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 4576 | . 2 ⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | |
2 | df-bj-ind 13962 | . . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)) | |
3 | 2 | bicomi 131 | . . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) ↔ Ind 𝑥) |
4 | 3 | abbii 2286 | . . 3 ⊢ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = {𝑥 ∣ Ind 𝑥} |
5 | 4 | inteqi 3835 | . 2 ⊢ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = ∩ {𝑥 ∣ Ind 𝑥} |
6 | 1, 5 | eqtri 2191 | 1 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ∈ wcel 2141 {cab 2156 ∀wral 2448 ∅c0 3414 ∩ cint 3831 suc csuc 4350 ωcom 4574 Ind wind 13961 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-int 3832 df-iom 4575 df-bj-ind 13962 |
This theorem is referenced by: bj-omind 13969 bj-omssind 13970 bj-ssom 13971 |
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