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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-indint | GIF version |
Description: The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.) |
Ref | Expression |
---|---|
bj-indint | ⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-ind 12710 | . . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)) | |
2 | 1 | simplbi 270 | . . . 4 ⊢ (Ind 𝑥 → ∅ ∈ 𝑥) |
3 | 2 | rgenw 2446 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥) |
4 | 0ex 3995 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | elintrab 3730 | . . 3 ⊢ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)) |
6 | 3, 5 | mpbir 145 | . 2 ⊢ ∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} |
7 | bj-indsuc 12711 | . . . . . 6 ⊢ (Ind 𝑥 → (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) | |
8 | 7 | a2i 11 | . . . . 5 ⊢ ((Ind 𝑥 → 𝑦 ∈ 𝑥) → (Ind 𝑥 → suc 𝑦 ∈ 𝑥)) |
9 | 8 | ralimi 2454 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥)) |
10 | vex 2644 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 10 | elintrab 3730 | . . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥)) |
12 | 10 | bj-sucex 12702 | . . . . 5 ⊢ suc 𝑦 ∈ V |
13 | 12 | elintrab 3730 | . . . 4 ⊢ (suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥)) |
14 | 9, 11, 13 | 3imtr4i 200 | . . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} → suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}) |
15 | 14 | rgen 2444 | . 2 ⊢ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} |
16 | df-bj-ind 12710 | . 2 ⊢ (Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥})) | |
17 | 6, 15, 16 | mpbir2an 894 | 1 ⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1448 ∀wral 2375 {crab 2379 ∅c0 3310 ∩ cint 3718 suc csuc 4225 Ind wind 12709 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-nul 3994 ax-pr 4069 ax-un 4293 ax-bd0 12592 ax-bdor 12595 ax-bdex 12598 ax-bdeq 12599 ax-bdel 12600 ax-bdsb 12601 ax-bdsep 12663 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-nul 3311 df-sn 3480 df-pr 3481 df-uni 3684 df-int 3719 df-suc 4231 df-bdc 12620 df-bj-ind 12710 |
This theorem is referenced by: bj-omind 12717 |
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