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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-indind | Unicode version |
Description: If is inductive and is "inductive in ", then is inductive. (Contributed by BJ, 25-Oct-2020.) |
Ref | Expression |
---|---|
bj-indind | Ind Ind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-ind 10051 | . . . 4 Ind | |
2 | id 19 | . . . . 5 | |
3 | 2 | an4s 522 | . . . 4 |
4 | 1, 3 | sylanb 268 | . . 3 Ind |
5 | elin 3126 | . . . . 5 | |
6 | 5 | biimpri 124 | . . . 4 |
7 | r19.26 2441 | . . . . . . . 8 | |
8 | 7 | biimpri 124 | . . . . . . 7 |
9 | simpl 102 | . . . . . . . . 9 | |
10 | simpr 103 | . . . . . . . . 9 | |
11 | elin 3126 | . . . . . . . . . 10 | |
12 | 11 | biimpri 124 | . . . . . . . . 9 |
13 | 9, 10, 12 | syl6an 1323 | . . . . . . . 8 |
14 | 13 | ralimi 2384 | . . . . . . 7 |
15 | 8, 14 | syl 14 | . . . . . 6 |
16 | df-ral 2311 | . . . . . . 7 | |
17 | elin 3126 | . . . . . . . . 9 | |
18 | pm3.31 249 | . . . . . . . . 9 | |
19 | 17, 18 | syl5bi 141 | . . . . . . . 8 |
20 | 19 | alimi 1344 | . . . . . . 7 |
21 | 16, 20 | sylbi 114 | . . . . . 6 |
22 | 15, 21 | syl 14 | . . . . 5 |
23 | df-ral 2311 | . . . . 5 | |
24 | 22, 23 | sylibr 137 | . . . 4 |
25 | 6, 24 | anim12i 321 | . . 3 |
26 | 4, 25 | syl 14 | . 2 Ind |
27 | df-bj-ind 10051 | . 2 Ind | |
28 | 26, 27 | sylibr 137 | 1 Ind Ind |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wal 1241 wcel 1393 wral 2306 cin 2916 c0 3224 csuc 4102 Ind wind 10050 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-in 2924 df-bj-ind 10051 |
This theorem is referenced by: peano5set 10064 |
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