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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 4508 for a nonconstructive proof of the general case. See bdfind 13133 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 987 | . . 3 | |
2 | simp2 982 | . . . . . 6 | |
3 | df-ral 2419 | . . . . . . . 8 | |
4 | alral 2476 | . . . . . . . 8 | |
5 | 3, 4 | sylbi 120 | . . . . . . 7 |
6 | 5 | 3ad2ant3 1004 | . . . . . 6 |
7 | 2, 6 | jca 304 | . . . . 5 |
8 | 3anass 966 | . . . . . 6 | |
9 | 8 | biimpri 132 | . . . . 5 |
10 | 7, 9 | sylan2 284 | . . . 4 |
11 | speano5 13131 | . . . 4 | |
12 | 10, 11 | syl 14 | . . 3 |
13 | 1, 12 | eqssd 3109 | . 2 |
14 | 13 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wal 1329 wceq 1331 wcel 1480 wral 2414 wss 3066 c0 3358 csuc 4282 com 4499 |
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-bdan 13002 ax-bdor 13003 ax-bdex 13006 ax-bdeq 13007 ax-bdel 13008 ax-bdsb 13009 ax-bdsep 13071 ax-infvn 13128 |
This theorem depends on definitions: df-bi 116 df-3an 964 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-in 3072 df-ss 3079 df-nul 3359 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-suc 4288 df-iom 4500 df-bdc 13028 df-bj-ind 13114 |
This theorem is referenced by: bdfind 13133 |
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