![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-bdind | GIF version |
Description: Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.) |
Ref | Expression |
---|---|
bj-bdind | ⊢ BOUNDED Ind 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-bd0el 15078 | . . 3 ⊢ BOUNDED ∅ ∈ 𝑥 | |
2 | bj-bdsucel 15092 | . . . 4 ⊢ BOUNDED suc 𝑦 ∈ 𝑥 | |
3 | 2 | ax-bdal 15028 | . . 3 ⊢ BOUNDED ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 |
4 | 1, 3 | ax-bdan 15025 | . 2 ⊢ BOUNDED (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) |
5 | df-bj-ind 15137 | . 2 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)) | |
6 | 4, 5 | bd0r 15035 | 1 ⊢ BOUNDED Ind 𝑥 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∈ wcel 2160 ∀wral 2468 ∅c0 3437 suc csuc 4383 BOUNDED wbd 15022 Ind wind 15136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-bd0 15023 ax-bdim 15024 ax-bdan 15025 ax-bdor 15026 ax-bdn 15027 ax-bdal 15028 ax-bdex 15029 ax-bdeq 15030 ax-bdel 15031 ax-bdsb 15032 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-sn 3613 df-suc 4389 df-bdc 15051 df-bj-ind 15137 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |