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Description: Bounded induction (principle of induction when is assumed to be a set) allowing a proof from basic constructive axioms. See find 4340 for a nonconstructive proof of the general case. See bdfind 10741 for a proof when is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
findset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 944 | . . 3 | |
2 | simp2 939 | . . . . . 6 | |
3 | df-ral 2353 | . . . . . . . 8 | |
4 | alral 2409 | . . . . . . . 8 | |
5 | 3, 4 | sylbi 119 | . . . . . . 7 |
6 | 5 | 3ad2ant3 961 | . . . . . 6 |
7 | 2, 6 | jca 300 | . . . . 5 |
8 | 3anass 923 | . . . . . 6 | |
9 | 8 | biimpri 131 | . . . . 5 |
10 | 7, 9 | sylan2 280 | . . . 4 |
11 | speano5 10739 | . . . 4 | |
12 | 10, 11 | syl 14 | . . 3 |
13 | 1, 12 | eqssd 3016 | . 2 |
14 | 13 | ex 113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wal 1282 wceq 1284 wcel 1433 wral 2348 wss 2973 c0 3251 csuc 4120 com 4331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-nul 3904 ax-pr 3964 ax-un 4188 ax-bd0 10604 ax-bdan 10606 ax-bdor 10607 ax-bdex 10610 ax-bdeq 10611 ax-bdel 10612 ax-bdsb 10613 ax-bdsep 10675 ax-infvn 10736 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-suc 4126 df-iom 4332 df-bdc 10632 df-bj-ind 10722 |
This theorem is referenced by: bdfind 10741 |
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