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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 6262). 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 1508 | . . . 4 | |
2 | nfra1 2466 | . . . 4 | |
3 | 1, 2 | nfan 1544 | . . 3 |
4 | nfv 1508 | . . . . . 6 | |
5 | 3, 4 | nfim 1551 | . . . . 5 |
6 | fveq2 5421 | . . . . . . 7 | |
7 | fveq2 5421 | . . . . . . 7 | |
8 | 6, 7 | eqeq12d 2154 | . . . . . 6 |
9 | 8 | imbi2d 229 | . . . . 5 |
10 | r19.21v 2509 | . . . . . 6 | |
11 | rsp 2480 | . . . . . . . . . 10 | |
12 | onss 4409 | . . . . . . . . . . . . . . . . . . 19 | |
13 | tfri3.1 | . . . . . . . . . . . . . . . . . . . . . 22 recs | |
14 | tfri3.2 | . . . . . . . . . . . . . . . . . . . . . 22 | |
15 | 13, 14 | tfri1 6262 | . . . . . . . . . . . . . . . . . . . . 21 |
16 | fvreseq 5524 | . . . . . . . . . . . . . . . . . . . . 21 | |
17 | 15, 16 | mpanl2 431 | . . . . . . . . . . . . . . . . . . . 20 |
18 | fveq2 5421 | . . . . . . . . . . . . . . . . . . . 20 | |
19 | 17, 18 | syl6bir 163 | . . . . . . . . . . . . . . . . . . 19 |
20 | 12, 19 | sylan2 284 | . . . . . . . . . . . . . . . . . 18 |
21 | 20 | ancoms 266 | . . . . . . . . . . . . . . . . 17 |
22 | 21 | imp 123 | . . . . . . . . . . . . . . . 16 |
23 | 22 | adantr 274 | . . . . . . . . . . . . . . 15 |
24 | 13, 14 | tfri2 6263 | . . . . . . . . . . . . . . . . . . . 20 |
25 | 24 | jctr 313 | . . . . . . . . . . . . . . . . . . 19 |
26 | jcab 592 | . . . . . . . . . . . . . . . . . . 19 | |
27 | 25, 26 | sylibr 133 | . . . . . . . . . . . . . . . . . 18 |
28 | eqeq12 2152 | . . . . . . . . . . . . . . . . . 18 | |
29 | 27, 28 | syl6 33 | . . . . . . . . . . . . . . . . 17 |
30 | 29 | imp 123 | . . . . . . . . . . . . . . . 16 |
31 | 30 | adantl 275 | . . . . . . . . . . . . . . 15 |
32 | 23, 31 | mpbird 166 | . . . . . . . . . . . . . 14 |
33 | 32 | exp43 369 | . . . . . . . . . . . . 13 |
34 | 33 | com4t 85 | . . . . . . . . . . . 12 |
35 | 34 | exp4a 363 | . . . . . . . . . . 11 |
36 | 35 | pm2.43d 50 | . . . . . . . . . 10 |
37 | 11, 36 | syl 14 | . . . . . . . . 9 |
38 | 37 | com3l 81 | . . . . . . . 8 |
39 | 38 | impd 252 | . . . . . . 7 |
40 | 39 | a2d 26 | . . . . . 6 |
41 | 10, 40 | syl5bi 151 | . . . . 5 |
42 | 5, 9, 41 | tfis2f 4498 | . . . 4 |
43 | 42 | com12 30 | . . 3 |
44 | 3, 43 | ralrimi 2503 | . 2 |
45 | eqfnfv 5518 | . . . 4 | |
46 | 15, 45 | mpan2 421 | . . 3 |
47 | 46 | biimpar 295 | . 2 |
48 | 44, 47 | syldan 280 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 cvv 2686 wss 3071 con0 4285 cres 4541 wfun 5117 wfn 5118 cfv 5123 recscrecs 6201 |
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-in1 603 ax-in2 604 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-recs 6202 |
This theorem is referenced by: (None) |
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