![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 9386 | . . . . 5 ⊢ (Ind x ↔ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)) | |
2 | 1 | simplbi 259 | . . . 4 ⊢ (Ind x → ∅ ∈ x) |
3 | 2 | rgenw 2370 | . . 3 ⊢ ∀x ∈ A (Ind x → ∅ ∈ x) |
4 | 0ex 3875 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | elintrab 3618 | . . 3 ⊢ (∅ ∈ ∩ {x ∈ A ∣ Ind x} ↔ ∀x ∈ A (Ind x → ∅ ∈ x)) |
6 | 3, 5 | mpbir 134 | . 2 ⊢ ∅ ∈ ∩ {x ∈ A ∣ Ind x} |
7 | bj-indsuc 9387 | . . . . . 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 2378 | . . . 4 ⊢ (∀x ∈ A (Ind x → y ∈ x) → ∀x ∈ A (Ind x → suc y ∈ x)) |
10 | vex 2554 | . . . . 5 ⊢ y ∈ V | |
11 | 10 | elintrab 3618 | . . . 4 ⊢ (y ∈ ∩ {x ∈ A ∣ Ind x} ↔ ∀x ∈ A (Ind x → y ∈ x)) |
12 | 10 | bj-sucex 9378 | . . . . 5 ⊢ suc y ∈ V |
13 | 12 | elintrab 3618 | . . . 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 2368 | . 2 ⊢ ∀y ∈ ∩ {x ∈ A ∣ Ind x}suc y ∈ ∩ {x ∈ A ∣ Ind x} |
16 | df-bj-ind 9386 | . 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 848 | 1 ⊢ Ind ∩ {x ∈ A ∣ Ind x} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1390 ∀wral 2300 {crab 2304 ∅c0 3218 ∩ cint 3606 suc csuc 4068 Ind wind 9385 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-nul 3874 ax-pr 3935 ax-un 4136 ax-bd0 9268 ax-bdor 9271 ax-bdex 9274 ax-bdeq 9275 ax-bdel 9276 ax-bdsb 9277 ax-bdsep 9339 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-nul 3219 df-sn 3373 df-pr 3374 df-uni 3572 df-int 3607 df-suc 4074 df-bdc 9296 df-bj-ind 9386 |
This theorem is referenced by: bj-omind 9393 |
Copyright terms: Public domain | W3C validator |