Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2392 | . . . . . . 7 |
3 | nfv 1421 | . . . . . . . 8 | |
4 | nfv 1421 | . . . . . . . . 9 | |
5 | nfs1v 1815 | . . . . . . . . 9 | |
6 | 4, 5 | nfim 1464 | . . . . . . . 8 |
7 | breq2 3768 | . . . . . . . . . . 11 | |
8 | 7 | imbi1d 220 | . . . . . . . . . 10 |
9 | 8 | ralbidv 2326 | . . . . . . . . 9 |
10 | sbequ12 1654 | . . . . . . . . 9 | |
11 | 9, 10 | imbi12d 223 | . . . . . . . 8 |
12 | 3, 6, 11 | cbvral 2529 | . . . . . . 7 |
13 | 2, 12 | sylib 127 | . . . . . 6 |
14 | frind.sb | . . . . . . . . . . . 12 | |
15 | 14 | elrab3 2699 | . . . . . . . . . . 11 |
16 | 15 | imbi2d 219 | . . . . . . . . . 10 |
17 | 16 | ralbiia 2338 | . . . . . . . . 9 |
18 | 17 | a1i 9 | . . . . . . . 8 |
19 | nfcv 2178 | . . . . . . . . . 10 | |
20 | nfcv 2178 | . . . . . . . . . 10 | |
21 | 19, 20, 5, 10 | elrabf 2696 | . . . . . . . . 9 |
22 | 21 | baib 828 | . . . . . . . 8 |
23 | 18, 22 | imbi12d 223 | . . . . . . 7 |
24 | 23 | ralbiia 2338 | . . . . . 6 |
25 | 13, 24 | sylibr 137 | . . . . 5 |
26 | frind.fr | . . . . . . . 8 | |
27 | df-frind 4069 | . . . . . . . 8 FrFor | |
28 | 26, 27 | sylib 127 | . . . . . . 7 FrFor |
29 | frind.a | . . . . . . . 8 | |
30 | rabexg 3900 | . . . . . . . 8 | |
31 | frforeq3 4084 | . . . . . . . . 9 FrFor FrFor | |
32 | 31 | spcgv 2640 | . . . . . . . 8 FrFor FrFor |
33 | 29, 30, 32 | 3syl 17 | . . . . . . 7 FrFor FrFor |
34 | 28, 33 | mpd 13 | . . . . . 6 FrFor |
35 | df-frfor 4068 | . . . . . 6 FrFor | |
36 | 34, 35 | sylib 127 | . . . . 5 |
37 | 25, 36 | mpd 13 | . . . 4 |
38 | ssrab 3018 | . . . 4 | |
39 | 37, 38 | sylib 127 | . . 3 |
40 | 39 | simprd 107 | . 2 |
41 | 40 | r19.21bi 2407 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wcel 1393 wsb 1645 wral 2306 crab 2310 cvv 2557 wss 2917 class class class wbr 3764 FrFor wfrfor 4064 wfr 4065 |
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 ax-sep 3875 |
This theorem depends on definitions: df-bi 110 df-3an 887 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-rab 2315 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-frfor 4068 df-frind 4069 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |