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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 4552 | . 2 ⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | |
2 | df-bj-ind 13544 | . . . . 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 2273 | . . 3 ⊢ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = {𝑥 ∣ Ind 𝑥} |
5 | 4 | inteqi 3812 | . 2 ⊢ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = ∩ {𝑥 ∣ Ind 𝑥} |
6 | 1, 5 | eqtri 2178 | 1 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1335 ∈ wcel 2128 {cab 2143 ∀wral 2435 ∅c0 3394 ∩ cint 3808 suc csuc 4326 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-int 3809 df-iom 4551 df-bj-ind 13544 |
This theorem is referenced by: bj-omind 13551 bj-omssind 13552 bj-ssom 13553 |
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