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Mirrors > Home > ILE Home > Th. List > tfri3 | Unicode version |
Description: Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5974). Finally, we show that is unique. We do this by showing that any class with the same properties of that we showed in parts 1 and 2 is identical to . (Contributed by Jim Kingdon, 4-May-2019.) |
Ref | Expression |
---|---|
tfri3.1 | recs |
tfri3.2 |
Ref | Expression |
---|---|
tfri3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1461 | . . . 4 | |
2 | nfra1 2397 | . . . 4 | |
3 | 1, 2 | nfan 1497 | . . 3 |
4 | nfv 1461 | . . . . . 6 | |
5 | 3, 4 | nfim 1504 | . . . . 5 |
6 | fveq2 5198 | . . . . . . 7 | |
7 | fveq2 5198 | . . . . . . 7 | |
8 | 6, 7 | eqeq12d 2095 | . . . . . 6 |
9 | 8 | imbi2d 228 | . . . . 5 |
10 | r19.21v 2438 | . . . . . 6 | |
11 | rsp 2411 | . . . . . . . . . 10 | |
12 | onss 4237 | . . . . . . . . . . . . . . . . . . 19 | |
13 | tfri3.1 | . . . . . . . . . . . . . . . . . . . . . 22 recs | |
14 | tfri3.2 | . . . . . . . . . . . . . . . . . . . . . 22 | |
15 | 13, 14 | tfri1 5974 | . . . . . . . . . . . . . . . . . . . . 21 |
16 | fvreseq 5292 | . . . . . . . . . . . . . . . . . . . . 21 | |
17 | 15, 16 | mpanl2 425 | . . . . . . . . . . . . . . . . . . . 20 |
18 | fveq2 5198 | . . . . . . . . . . . . . . . . . . . 20 | |
19 | 17, 18 | syl6bir 162 | . . . . . . . . . . . . . . . . . . 19 |
20 | 12, 19 | sylan2 280 | . . . . . . . . . . . . . . . . . 18 |
21 | 20 | ancoms 264 | . . . . . . . . . . . . . . . . 17 |
22 | 21 | imp 122 | . . . . . . . . . . . . . . . 16 |
23 | 22 | adantr 270 | . . . . . . . . . . . . . . 15 |
24 | 13, 14 | tfri2 5975 | . . . . . . . . . . . . . . . . . . . 20 |
25 | 24 | jctr 308 | . . . . . . . . . . . . . . . . . . 19 |
26 | jcab 567 | . . . . . . . . . . . . . . . . . . 19 | |
27 | 25, 26 | sylibr 132 | . . . . . . . . . . . . . . . . . 18 |
28 | eqeq12 2093 | . . . . . . . . . . . . . . . . . 18 | |
29 | 27, 28 | syl6 33 | . . . . . . . . . . . . . . . . 17 |
30 | 29 | imp 122 | . . . . . . . . . . . . . . . 16 |
31 | 30 | adantl 271 | . . . . . . . . . . . . . . 15 |
32 | 23, 31 | mpbird 165 | . . . . . . . . . . . . . 14 |
33 | 32 | exp43 364 | . . . . . . . . . . . . 13 |
34 | 33 | com4t 84 | . . . . . . . . . . . 12 |
35 | 34 | exp4a 358 | . . . . . . . . . . 11 |
36 | 35 | pm2.43d 49 | . . . . . . . . . 10 |
37 | 11, 36 | syl 14 | . . . . . . . . 9 |
38 | 37 | com3l 80 | . . . . . . . 8 |
39 | 38 | impd 251 | . . . . . . 7 |
40 | 39 | a2d 26 | . . . . . 6 |
41 | 10, 40 | syl5bi 150 | . . . . 5 |
42 | 5, 9, 41 | tfis2f 4325 | . . . 4 |
43 | 42 | com12 30 | . . 3 |
44 | 3, 43 | ralrimi 2432 | . 2 |
45 | eqfnfv 5286 | . . . 4 | |
46 | 15, 45 | mpan2 415 | . . 3 |
47 | 46 | biimpar 291 | . 2 |
48 | 44, 47 | syldan 276 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 cvv 2601 wss 2973 con0 4118 cres 4365 wfun 4916 wfn 4917 cfv 4922 recscrecs 5942 |
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-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-recs 5943 |
This theorem is referenced by: (None) |
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