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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 13544 | . . . . 5 Ind | |
2 | 1 | simplbi 272 | . . . 4 Ind |
3 | 2 | rgenw 2512 | . . 3 Ind |
4 | 0ex 4092 | . . . 4 | |
5 | 4 | elintrab 3820 | . . 3 Ind Ind |
6 | 3, 5 | mpbir 145 | . 2 Ind |
7 | bj-indsuc 13545 | . . . . . 6 Ind | |
8 | 7 | a2i 11 | . . . . 5 Ind Ind |
9 | 8 | ralimi 2520 | . . . 4 Ind Ind |
10 | vex 2715 | . . . . 5 | |
11 | 10 | elintrab 3820 | . . . 4 Ind Ind |
12 | 10 | bj-sucex 13540 | . . . . 5 |
13 | 12 | elintrab 3820 | . . . 4 Ind Ind |
14 | 9, 11, 13 | 3imtr4i 200 | . . 3 Ind Ind |
15 | 14 | rgen 2510 | . 2 Ind Ind |
16 | df-bj-ind 13544 | . 2 Ind Ind Ind Ind Ind | |
17 | 6, 15, 16 | mpbir2an 927 | 1 Ind Ind |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2128 wral 2435 crab 2439 c0 3394 cint 3808 csuc 4326 Ind wind 13543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-nul 4091 ax-pr 4170 ax-un 4394 ax-bd0 13430 ax-bdor 13433 ax-bdex 13436 ax-bdeq 13437 ax-bdel 13438 ax-bdsb 13439 ax-bdsep 13501 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-nul 3395 df-sn 3566 df-pr 3567 df-uni 3774 df-int 3809 df-suc 4332 df-bdc 13458 df-bj-ind 13544 |
This theorem is referenced by: bj-omind 13551 |
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