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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-omssind | GIF version |
Description: ω is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-omssind | ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2281 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1508 | . . 3 ⊢ Ⅎ𝑥Ind 𝐴 | |
3 | bj-indeq 13127 | . . . 4 ⊢ (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴)) | |
4 | 3 | biimprd 157 | . . 3 ⊢ (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥)) |
5 | 1, 2, 4 | bj-intabssel1 12997 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)) |
6 | bj-dfom 13131 | . . 3 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥} | |
7 | 6 | sseq1i 3123 | . 2 ⊢ (ω ⊆ 𝐴 ↔ ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴) |
8 | 5, 7 | syl6ibr 161 | 1 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 {cab 2125 ⊆ wss 3071 ∩ cint 3771 ωcom 4504 Ind wind 13124 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-in 3077 df-ss 3084 df-int 3772 df-iom 4505 df-bj-ind 13125 |
This theorem is referenced by: bj-om 13135 peano5set 13138 |
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