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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 5951). 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 1421 | . . . 4 | |
2 | nfra1 2355 | . . . 4 | |
3 | 1, 2 | nfan 1457 | . . 3 |
4 | nfv 1421 | . . . . . 6 | |
5 | 3, 4 | nfim 1464 | . . . . 5 |
6 | fveq2 5178 | . . . . . . 7 | |
7 | fveq2 5178 | . . . . . . 7 | |
8 | 6, 7 | eqeq12d 2054 | . . . . . 6 |
9 | 8 | imbi2d 219 | . . . . 5 |
10 | r19.21v 2396 | . . . . . 6 | |
11 | rsp 2369 | . . . . . . . . . 10 | |
12 | onss 4219 | . . . . . . . . . . . . . . . . . . 19 | |
13 | tfri3.1 | . . . . . . . . . . . . . . . . . . . . . 22 recs | |
14 | tfri3.2 | . . . . . . . . . . . . . . . . . . . . . 22 | |
15 | 13, 14 | tfri1 5951 | . . . . . . . . . . . . . . . . . . . . 21 |
16 | fvreseq 5271 | . . . . . . . . . . . . . . . . . . . . 21 | |
17 | 15, 16 | mpanl2 411 | . . . . . . . . . . . . . . . . . . . 20 |
18 | fveq2 5178 | . . . . . . . . . . . . . . . . . . . 20 | |
19 | 17, 18 | syl6bir 153 | . . . . . . . . . . . . . . . . . . 19 |
20 | 12, 19 | sylan2 270 | . . . . . . . . . . . . . . . . . 18 |
21 | 20 | ancoms 255 | . . . . . . . . . . . . . . . . 17 |
22 | 21 | imp 115 | . . . . . . . . . . . . . . . 16 |
23 | 22 | adantr 261 | . . . . . . . . . . . . . . 15 |
24 | 13, 14 | tfri2 5952 | . . . . . . . . . . . . . . . . . . . 20 |
25 | 24 | jctr 298 | . . . . . . . . . . . . . . . . . . 19 |
26 | jcab 535 | . . . . . . . . . . . . . . . . . . 19 | |
27 | 25, 26 | sylibr 137 | . . . . . . . . . . . . . . . . . 18 |
28 | eqeq12 2052 | . . . . . . . . . . . . . . . . . 18 | |
29 | 27, 28 | syl6 29 | . . . . . . . . . . . . . . . . 17 |
30 | 29 | imp 115 | . . . . . . . . . . . . . . . 16 |
31 | 30 | adantl 262 | . . . . . . . . . . . . . . 15 |
32 | 23, 31 | mpbird 156 | . . . . . . . . . . . . . 14 |
33 | 32 | exp43 354 | . . . . . . . . . . . . 13 |
34 | 33 | com4t 79 | . . . . . . . . . . . 12 |
35 | 34 | exp4a 348 | . . . . . . . . . . 11 |
36 | 35 | pm2.43d 44 | . . . . . . . . . 10 |
37 | 11, 36 | syl 14 | . . . . . . . . 9 |
38 | 37 | com3l 75 | . . . . . . . 8 |
39 | 38 | impd 242 | . . . . . . 7 |
40 | 39 | a2d 23 | . . . . . 6 |
41 | 10, 40 | syl5bi 141 | . . . . 5 |
42 | 5, 9, 41 | tfis2f 4307 | . . . 4 |
43 | 42 | com12 27 | . . 3 |
44 | 3, 43 | ralrimi 2390 | . 2 |
45 | eqfnfv 5265 | . . . 4 | |
46 | 15, 45 | mpan2 401 | . . 3 |
47 | 46 | biimpar 281 | . 2 |
48 | 44, 47 | syldan 266 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 cvv 2557 wss 2917 con0 4100 cres 4347 wfun 4896 wfn 4897 cfv 4902 recscrecs 5919 |
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-in1 544 ax-in2 545 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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-recs 5920 |
This theorem is referenced by: (None) |
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