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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 2178 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1421 | . . 3 ⊢ Ⅎ𝑥Ind 𝐴 | |
3 | bj-indeq 10053 | . . . 4 ⊢ (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴)) | |
4 | 3 | biimprd 147 | . . 3 ⊢ (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥)) |
5 | 1, 2, 4 | bj-intabssel1 9929 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)) |
6 | bj-dfom 10057 | . . 3 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥} | |
7 | 6 | sseq1i 2969 | . 2 ⊢ (ω ⊆ 𝐴 ↔ ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴) |
8 | 5, 7 | syl6ibr 151 | 1 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 {cab 2026 ⊆ wss 2917 ∩ cint 3615 ωcom 4313 Ind wind 10050 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-in 2924 df-ss 2931 df-int 3616 df-iom 4314 df-bj-ind 10051 |
This theorem is referenced by: bj-om 10061 peano5set 10064 |
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