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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 13052 | . . . 4 Ind | |
2 | id 19 | . . . . 5 | |
3 | 2 | an4s 562 | . . . 4 |
4 | 1, 3 | sylanb 282 | . . 3 Ind |
5 | elin 3229 | . . . . 5 | |
6 | 5 | biimpri 132 | . . . 4 |
7 | r19.26 2535 | . . . . . . . 8 | |
8 | 7 | biimpri 132 | . . . . . . 7 |
9 | simpl 108 | . . . . . . . . 9 | |
10 | simpr 109 | . . . . . . . . 9 | |
11 | elin 3229 | . . . . . . . . . 10 | |
12 | 11 | biimpri 132 | . . . . . . . . 9 |
13 | 9, 10, 12 | syl6an 1395 | . . . . . . . 8 |
14 | 13 | ralimi 2472 | . . . . . . 7 |
15 | 8, 14 | syl 14 | . . . . . 6 |
16 | df-ral 2398 | . . . . . . 7 | |
17 | elin 3229 | . . . . . . . . 9 | |
18 | pm3.31 260 | . . . . . . . . 9 | |
19 | 17, 18 | syl5bi 151 | . . . . . . . 8 |
20 | 19 | alimi 1416 | . . . . . . 7 |
21 | 16, 20 | sylbi 120 | . . . . . 6 |
22 | 15, 21 | syl 14 | . . . . 5 |
23 | df-ral 2398 | . . . . 5 | |
24 | 22, 23 | sylibr 133 | . . . 4 |
25 | 6, 24 | anim12i 336 | . . 3 |
26 | 4, 25 | syl 14 | . 2 Ind |
27 | df-bj-ind 13052 | . 2 Ind | |
28 | 26, 27 | sylibr 133 | 1 Ind Ind |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1314 wcel 1465 wral 2393 cin 3040 c0 3333 csuc 4257 Ind wind 13051 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-in 3047 df-bj-ind 13052 |
This theorem is referenced by: peano5set 13065 |
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