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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-indint | Unicode version |
Description: The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.) |
Ref | Expression |
---|---|
bj-indint | Ind Ind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-ind 10722 | . . . . 5 Ind | |
2 | 1 | simplbi 268 | . . . 4 Ind |
3 | 2 | rgenw 2418 | . . 3 Ind |
4 | 0ex 3905 | . . . 4 | |
5 | 4 | elintrab 3648 | . . 3 Ind Ind |
6 | 3, 5 | mpbir 144 | . 2 Ind |
7 | bj-indsuc 10723 | . . . . . 6 Ind | |
8 | 7 | a2i 11 | . . . . 5 Ind Ind |
9 | 8 | ralimi 2426 | . . . 4 Ind Ind |
10 | vex 2604 | . . . . 5 | |
11 | 10 | elintrab 3648 | . . . 4 Ind Ind |
12 | 10 | bj-sucex 10714 | . . . . 5 |
13 | 12 | elintrab 3648 | . . . 4 Ind Ind |
14 | 9, 11, 13 | 3imtr4i 199 | . . 3 Ind Ind |
15 | 14 | rgen 2416 | . 2 Ind Ind |
16 | df-bj-ind 10722 | . 2 Ind Ind Ind Ind Ind | |
17 | 6, 15, 16 | mpbir2an 883 | 1 Ind Ind |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1433 wral 2348 crab 2352 c0 3251 cint 3636 csuc 4120 Ind wind 10721 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-nul 3904 ax-pr 3964 ax-un 4188 ax-bd0 10604 ax-bdor 10607 ax-bdex 10610 ax-bdeq 10611 ax-bdel 10612 ax-bdsb 10613 ax-bdsep 10675 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-nul 3252 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-suc 4126 df-bdc 10632 df-bj-ind 10722 |
This theorem is referenced by: bj-omind 10729 |
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