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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 6324). 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 1515 | . . . 4 | |
2 | nfra1 2495 | . . . 4 | |
3 | 1, 2 | nfan 1552 | . . 3 |
4 | nfv 1515 | . . . . . 6 | |
5 | 3, 4 | nfim 1559 | . . . . 5 |
6 | fveq2 5480 | . . . . . . 7 | |
7 | fveq2 5480 | . . . . . . 7 | |
8 | 6, 7 | eqeq12d 2179 | . . . . . 6 |
9 | 8 | imbi2d 229 | . . . . 5 |
10 | r19.21v 2541 | . . . . . 6 | |
11 | rsp 2511 | . . . . . . . . . 10 | |
12 | onss 4464 | . . . . . . . . . . . . . . . . . . 19 | |
13 | tfri3.1 | . . . . . . . . . . . . . . . . . . . . . 22 recs | |
14 | tfri3.2 | . . . . . . . . . . . . . . . . . . . . . 22 | |
15 | 13, 14 | tfri1 6324 | . . . . . . . . . . . . . . . . . . . . 21 |
16 | fvreseq 5583 | . . . . . . . . . . . . . . . . . . . . 21 | |
17 | 15, 16 | mpanl2 432 | . . . . . . . . . . . . . . . . . . . 20 |
18 | fveq2 5480 | . . . . . . . . . . . . . . . . . . . 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 6325 | . . . . . . . . . . . . . . . . . . . 20 |
25 | 24 | jctr 313 | . . . . . . . . . . . . . . . . . . 19 |
26 | jcab 593 | . . . . . . . . . . . . . . . . . . 19 | |
27 | 25, 26 | sylibr 133 | . . . . . . . . . . . . . . . . . 18 |
28 | eqeq12 2177 | . . . . . . . . . . . . . . . . . 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 370 | . . . . . . . . . . . . 13 |
34 | 33 | com4t 85 | . . . . . . . . . . . 12 |
35 | 34 | exp4a 364 | . . . . . . . . . . 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 4555 | . . . 4 |
43 | 42 | com12 30 | . . 3 |
44 | 3, 43 | ralrimi 2535 | . 2 |
45 | eqfnfv 5577 | . . . 4 | |
46 | 15, 45 | mpan2 422 | . . 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 1342 wcel 2135 wral 2442 cvv 2721 wss 3111 con0 4335 cres 4600 wfun 5176 wfn 5177 cfv 5182 recscrecs 6263 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-recs 6264 |
This theorem is referenced by: (None) |
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