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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 10051 | . . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)) | |
2 | 1 | simplbi 259 | . . . 4 ⊢ (Ind 𝑥 → ∅ ∈ 𝑥) |
3 | 2 | rgenw 2376 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥) |
4 | 0ex 3884 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | elintrab 3627 | . . 3 ⊢ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)) |
6 | 3, 5 | mpbir 134 | . 2 ⊢ ∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} |
7 | bj-indsuc 10052 | . . . . . 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 2384 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥)) |
10 | vex 2560 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 10 | elintrab 3627 | . . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥)) |
12 | 10 | bj-sucex 10043 | . . . . 5 ⊢ suc 𝑦 ∈ V |
13 | 12 | elintrab 3627 | . . . 4 ⊢ (suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥)) |
14 | 9, 11, 13 | 3imtr4i 190 | . . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} → suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}) |
15 | 14 | rgen 2374 | . 2 ⊢ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} |
16 | df-bj-ind 10051 | . 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 849 | 1 ⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ∀wral 2306 {crab 2310 ∅c0 3224 ∩ cint 3615 suc csuc 4102 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-in1 544 ax-in2 545 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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 ax-pr 3944 ax-un 4170 ax-bd0 9933 ax-bdor 9936 ax-bdex 9939 ax-bdeq 9940 ax-bdel 9941 ax-bdsb 9942 ax-bdsep 10004 |
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-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-nul 3225 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-bdc 9961 df-bj-ind 10051 |
This theorem is referenced by: bj-omind 10058 |
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