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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 9988 | . . 3 ⊢ BOUNDED ∅ ∈ 𝑥 | |
2 | bj-bdsucel 10002 | . . . 4 ⊢ BOUNDED suc 𝑦 ∈ 𝑥 | |
3 | 2 | ax-bdal 9938 | . . 3 ⊢ BOUNDED ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 |
4 | 1, 3 | ax-bdan 9935 | . 2 ⊢ BOUNDED (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) |
5 | df-bj-ind 10051 | . 2 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)) | |
6 | 4, 5 | bd0r 9945 | 1 ⊢ BOUNDED Ind 𝑥 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∈ wcel 1393 ∀wral 2306 ∅c0 3224 suc csuc 4102 BOUNDED wbd 9932 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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-bd0 9933 ax-bdim 9934 ax-bdan 9935 ax-bdor 9936 ax-bdn 9937 ax-bdal 9938 ax-bdex 9939 ax-bdeq 9940 ax-bdel 9941 ax-bdsb 9942 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 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-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-suc 4108 df-bdc 9961 df-bj-ind 10051 |
This theorem is referenced by: (None) |
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