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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 13962 | . . . . 5 Ind | |
2 | 1 | simplbi 272 | . . . 4 Ind |
3 | 2 | rgenw 2525 | . . 3 Ind |
4 | 0ex 4116 | . . . 4 | |
5 | 4 | elintrab 3843 | . . 3 Ind Ind |
6 | 3, 5 | mpbir 145 | . 2 Ind |
7 | bj-indsuc 13963 | . . . . . 6 Ind | |
8 | 7 | a2i 11 | . . . . 5 Ind Ind |
9 | 8 | ralimi 2533 | . . . 4 Ind Ind |
10 | vex 2733 | . . . . 5 | |
11 | 10 | elintrab 3843 | . . . 4 Ind Ind |
12 | 10 | bj-sucex 13958 | . . . . 5 |
13 | 12 | elintrab 3843 | . . . 4 Ind Ind |
14 | 9, 11, 13 | 3imtr4i 200 | . . 3 Ind Ind |
15 | 14 | rgen 2523 | . 2 Ind Ind |
16 | df-bj-ind 13962 | . 2 Ind Ind Ind Ind Ind | |
17 | 6, 15, 16 | mpbir2an 937 | 1 Ind Ind |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2141 wral 2448 crab 2452 c0 3414 cint 3831 csuc 4350 Ind wind 13961 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-nul 4115 ax-pr 4194 ax-un 4418 ax-bd0 13848 ax-bdor 13851 ax-bdex 13854 ax-bdeq 13855 ax-bdel 13856 ax-bdsb 13857 ax-bdsep 13919 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-nul 3415 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-suc 4356 df-bdc 13876 df-bj-ind 13962 |
This theorem is referenced by: bj-omind 13969 |
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