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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 2543 | . . . . . . 7 |
3 | nfv 1521 | . . . . . . . 8 | |
4 | nfv 1521 | . . . . . . . . 9 | |
5 | nfs1v 1932 | . . . . . . . . 9 | |
6 | 4, 5 | nfim 1565 | . . . . . . . 8 |
7 | breq2 3991 | . . . . . . . . . . 11 | |
8 | 7 | imbi1d 230 | . . . . . . . . . 10 |
9 | 8 | ralbidv 2470 | . . . . . . . . 9 |
10 | sbequ12 1764 | . . . . . . . . 9 | |
11 | 9, 10 | imbi12d 233 | . . . . . . . 8 |
12 | 3, 6, 11 | cbvral 2692 | . . . . . . 7 |
13 | 2, 12 | sylib 121 | . . . . . 6 |
14 | frind.sb | . . . . . . . . . . . 12 | |
15 | 14 | elrab3 2887 | . . . . . . . . . . 11 |
16 | 15 | imbi2d 229 | . . . . . . . . . 10 |
17 | 16 | ralbiia 2484 | . . . . . . . . 9 |
18 | 17 | a1i 9 | . . . . . . . 8 |
19 | nfcv 2312 | . . . . . . . . . 10 | |
20 | nfcv 2312 | . . . . . . . . . 10 | |
21 | 19, 20, 5, 10 | elrabf 2884 | . . . . . . . . 9 |
22 | 21 | baib 914 | . . . . . . . 8 |
23 | 18, 22 | imbi12d 233 | . . . . . . 7 |
24 | 23 | ralbiia 2484 | . . . . . 6 |
25 | 13, 24 | sylibr 133 | . . . . 5 |
26 | frind.fr | . . . . . . . 8 | |
27 | df-frind 4315 | . . . . . . . 8 FrFor | |
28 | 26, 27 | sylib 121 | . . . . . . 7 FrFor |
29 | frind.a | . . . . . . . 8 | |
30 | rabexg 4130 | . . . . . . . 8 | |
31 | frforeq3 4330 | . . . . . . . . 9 FrFor FrFor | |
32 | 31 | spcgv 2817 | . . . . . . . 8 FrFor FrFor |
33 | 29, 30, 32 | 3syl 17 | . . . . . . 7 FrFor FrFor |
34 | 28, 33 | mpd 13 | . . . . . 6 FrFor |
35 | df-frfor 4314 | . . . . . 6 FrFor | |
36 | 34, 35 | sylib 121 | . . . . 5 |
37 | 25, 36 | mpd 13 | . . . 4 |
38 | ssrab 3225 | . . . 4 | |
39 | 37, 38 | sylib 121 | . . 3 |
40 | 39 | simprd 113 | . 2 |
41 | 40 | r19.21bi 2558 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1346 wsb 1755 wcel 2141 wral 2448 crab 2452 cvv 2730 wss 3121 class class class wbr 3987 FrFor wfrfor 4310 wfr 4311 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4105 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-frfor 4314 df-frind 4315 |
This theorem is referenced by: (None) |
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