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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 13114 | . . . . 5 Ind | |
2 | 1 | simplbi 272 | . . . 4 Ind |
3 | 2 | rgenw 2485 | . . 3 Ind |
4 | 0ex 4050 | . . . 4 | |
5 | 4 | elintrab 3778 | . . 3 Ind Ind |
6 | 3, 5 | mpbir 145 | . 2 Ind |
7 | bj-indsuc 13115 | . . . . . 6 Ind | |
8 | 7 | a2i 11 | . . . . 5 Ind Ind |
9 | 8 | ralimi 2493 | . . . 4 Ind Ind |
10 | vex 2684 | . . . . 5 | |
11 | 10 | elintrab 3778 | . . . 4 Ind Ind |
12 | 10 | bj-sucex 13110 | . . . . 5 |
13 | 12 | elintrab 3778 | . . . 4 Ind Ind |
14 | 9, 11, 13 | 3imtr4i 200 | . . 3 Ind Ind |
15 | 14 | rgen 2483 | . 2 Ind Ind |
16 | df-bj-ind 13114 | . 2 Ind Ind Ind Ind Ind | |
17 | 6, 15, 16 | mpbir2an 926 | 1 Ind Ind |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 wral 2414 crab 2418 c0 3358 cint 3766 csuc 4282 Ind wind 13113 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-nul 4049 ax-pr 4126 ax-un 4350 ax-bd0 13000 ax-bdor 13003 ax-bdex 13006 ax-bdeq 13007 ax-bdel 13008 ax-bdsb 13009 ax-bdsep 13071 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-nul 3359 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-suc 4288 df-bdc 13028 df-bj-ind 13114 |
This theorem is referenced by: bj-omind 13121 |
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