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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 9323 | . . 3 ⊢ BOUNDED ∅ ∈ x | |
2 | bj-bdsucel 9337 | . . . 4 ⊢ BOUNDED suc y ∈ x | |
3 | 2 | ax-bdal 9273 | . . 3 ⊢ BOUNDED ∀y ∈ x suc y ∈ x |
4 | 1, 3 | ax-bdan 9270 | . 2 ⊢ BOUNDED (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x) |
5 | df-bj-ind 9386 | . 2 ⊢ (Ind x ↔ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)) | |
6 | 4, 5 | bd0r 9280 | 1 ⊢ BOUNDED Ind x |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∈ wcel 1390 ∀wral 2300 ∅c0 3218 suc csuc 4068 BOUNDED wbd 9267 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-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-bd0 9268 ax-bdim 9269 ax-bdan 9270 ax-bdor 9271 ax-bdn 9272 ax-bdal 9273 ax-bdex 9274 ax-bdeq 9275 ax-bdel 9276 ax-bdsb 9277 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-fal 1248 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-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-suc 4074 df-bdc 9296 df-bj-ind 9386 |
This theorem is referenced by: (None) |
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