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Mirrors > Home > ILE Home > Th. List > frind | Unicode version |
Description: Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.) |
Ref | Expression |
---|---|
frind.sb | |
frind.ind | |
frind.fr | |
frind.a |
Ref | Expression |
---|---|
frind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frind.ind | . . . . . . . 8 | |
2 | 1 | ralrimiva 2505 | . . . . . . 7 |
3 | nfv 1508 | . . . . . . . 8 | |
4 | nfv 1508 | . . . . . . . . 9 | |
5 | nfs1v 1912 | . . . . . . . . 9 | |
6 | 4, 5 | nfim 1551 | . . . . . . . 8 |
7 | breq2 3933 | . . . . . . . . . . 11 | |
8 | 7 | imbi1d 230 | . . . . . . . . . 10 |
9 | 8 | ralbidv 2437 | . . . . . . . . 9 |
10 | sbequ12 1744 | . . . . . . . . 9 | |
11 | 9, 10 | imbi12d 233 | . . . . . . . 8 |
12 | 3, 6, 11 | cbvral 2650 | . . . . . . 7 |
13 | 2, 12 | sylib 121 | . . . . . 6 |
14 | frind.sb | . . . . . . . . . . . 12 | |
15 | 14 | elrab3 2841 | . . . . . . . . . . 11 |
16 | 15 | imbi2d 229 | . . . . . . . . . 10 |
17 | 16 | ralbiia 2449 | . . . . . . . . 9 |
18 | 17 | a1i 9 | . . . . . . . 8 |
19 | nfcv 2281 | . . . . . . . . . 10 | |
20 | nfcv 2281 | . . . . . . . . . 10 | |
21 | 19, 20, 5, 10 | elrabf 2838 | . . . . . . . . 9 |
22 | 21 | baib 904 | . . . . . . . 8 |
23 | 18, 22 | imbi12d 233 | . . . . . . 7 |
24 | 23 | ralbiia 2449 | . . . . . 6 |
25 | 13, 24 | sylibr 133 | . . . . 5 |
26 | frind.fr | . . . . . . . 8 | |
27 | df-frind 4254 | . . . . . . . 8 FrFor | |
28 | 26, 27 | sylib 121 | . . . . . . 7 FrFor |
29 | frind.a | . . . . . . . 8 | |
30 | rabexg 4071 | . . . . . . . 8 | |
31 | frforeq3 4269 | . . . . . . . . 9 FrFor FrFor | |
32 | 31 | spcgv 2773 | . . . . . . . 8 FrFor FrFor |
33 | 29, 30, 32 | 3syl 17 | . . . . . . 7 FrFor FrFor |
34 | 28, 33 | mpd 13 | . . . . . 6 FrFor |
35 | df-frfor 4253 | . . . . . 6 FrFor | |
36 | 34, 35 | sylib 121 | . . . . 5 |
37 | 25, 36 | mpd 13 | . . . 4 |
38 | ssrab 3175 | . . . 4 | |
39 | 37, 38 | sylib 121 | . . 3 |
40 | 39 | simprd 113 | . 2 |
41 | 40 | r19.21bi 2520 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wcel 1480 wsb 1735 wral 2416 crab 2420 cvv 2686 wss 3071 class class class wbr 3929 FrFor wfrfor 4249 wfr 4250 |
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 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-frfor 4253 df-frind 4254 |
This theorem is referenced by: (None) |
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