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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 13644 | . . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)) | |
2 | 1 | simplbi 272 | . . . 4 ⊢ (Ind 𝑥 → ∅ ∈ 𝑥) |
3 | 2 | rgenw 2519 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥) |
4 | 0ex 4103 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | elintrab 3830 | . . 3 ⊢ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)) |
6 | 3, 5 | mpbir 145 | . 2 ⊢ ∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} |
7 | bj-indsuc 13645 | . . . . . 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 2527 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥)) |
10 | vex 2724 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 10 | elintrab 3830 | . . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥)) |
12 | 10 | bj-sucex 13640 | . . . . 5 ⊢ suc 𝑦 ∈ V |
13 | 12 | elintrab 3830 | . . . 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 2517 | . 2 ⊢ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} |
16 | df-bj-ind 13644 | . 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 931 | 1 ⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 ∀wral 2442 {crab 2446 ∅c0 3404 ∩ cint 3818 suc csuc 4337 Ind wind 13643 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-nul 4102 ax-pr 4181 ax-un 4405 ax-bd0 13530 ax-bdor 13533 ax-bdex 13536 ax-bdeq 13537 ax-bdel 13538 ax-bdsb 13539 ax-bdsep 13601 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-nul 3405 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-suc 4343 df-bdc 13558 df-bj-ind 13644 |
This theorem is referenced by: bj-omind 13651 |
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