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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 2258 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1493 | . . 3 ⊢ Ⅎ𝑥Ind 𝐴 | |
3 | bj-indeq 13054 | . . . 4 ⊢ (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴)) | |
4 | 3 | biimprd 157 | . . 3 ⊢ (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥)) |
5 | 1, 2, 4 | bj-intabssel1 12924 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)) |
6 | bj-dfom 13058 | . . 3 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥} | |
7 | 6 | sseq1i 3093 | . 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 1316 ∈ wcel 1465 {cab 2103 ⊆ 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 peano5set 13065 |
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