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Mirrors > Home > ILE Home > Th. List > tfr0dm | Unicode version |
Description: Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.) |
Ref | Expression |
---|---|
tfr.1 | recs |
Ref | Expression |
---|---|
tfr0dm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4050 | . . . . 5 | |
2 | opexg 4145 | . . . . 5 | |
3 | 1, 2 | mpan 420 | . . . 4 |
4 | snidg 3549 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | fnsng 5165 | . . . . 5 | |
7 | 1, 6 | mpan 420 | . . . 4 |
8 | fvsng 5609 | . . . . . . 7 | |
9 | 1, 8 | mpan 420 | . . . . . 6 |
10 | res0 4818 | . . . . . . 7 | |
11 | 10 | fveq2i 5417 | . . . . . 6 |
12 | 9, 11 | syl6eqr 2188 | . . . . 5 |
13 | fveq2 5414 | . . . . . . 7 | |
14 | reseq2 4809 | . . . . . . . 8 | |
15 | 14 | fveq2d 5418 | . . . . . . 7 |
16 | 13, 15 | eqeq12d 2152 | . . . . . 6 |
17 | 1, 16 | ralsn 3562 | . . . . 5 |
18 | 12, 17 | sylibr 133 | . . . 4 |
19 | suc0 4328 | . . . . . 6 | |
20 | 0elon 4309 | . . . . . . 7 | |
21 | 20 | onsuci 4427 | . . . . . 6 |
22 | 19, 21 | eqeltrri 2211 | . . . . 5 |
23 | fneq2 5207 | . . . . . . 7 | |
24 | raleq 2624 | . . . . . . 7 | |
25 | 23, 24 | anbi12d 464 | . . . . . 6 |
26 | 25 | rspcev 2784 | . . . . 5 |
27 | 22, 26 | mpan 420 | . . . 4 |
28 | 7, 18, 27 | syl2anc 408 | . . 3 |
29 | snexg 4103 | . . . . 5 | |
30 | eleq2 2201 | . . . . . . 7 | |
31 | fneq1 5206 | . . . . . . . . 9 | |
32 | fveq1 5413 | . . . . . . . . . . 11 | |
33 | reseq1 4808 | . . . . . . . . . . . 12 | |
34 | 33 | fveq2d 5418 | . . . . . . . . . . 11 |
35 | 32, 34 | eqeq12d 2152 | . . . . . . . . . 10 |
36 | 35 | ralbidv 2435 | . . . . . . . . 9 |
37 | 31, 36 | anbi12d 464 | . . . . . . . 8 |
38 | 37 | rexbidv 2436 | . . . . . . 7 |
39 | 30, 38 | anbi12d 464 | . . . . . 6 |
40 | 39 | spcegv 2769 | . . . . 5 |
41 | 3, 29, 40 | 3syl 17 | . . . 4 |
42 | tfr.1 | . . . . . 6 recs | |
43 | 42 | eleq2i 2204 | . . . . 5 recs |
44 | df-recs 6195 | . . . . . 6 recs | |
45 | 44 | eleq2i 2204 | . . . . 5 recs |
46 | eluniab 3743 | . . . . 5 | |
47 | 43, 45, 46 | 3bitri 205 | . . . 4 |
48 | 41, 47 | syl6ibr 161 | . . 3 |
49 | 5, 28, 48 | mp2and 429 | . 2 |
50 | opeldmg 4739 | . . 3 | |
51 | 1, 50 | mpan 420 | . 2 |
52 | 49, 51 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 cab 2123 wral 2414 wrex 2415 cvv 2681 c0 3358 csn 3522 cop 3525 cuni 3731 con0 4280 csuc 4282 cdm 4534 cres 4536 wfn 5113 cfv 5118 recscrecs 6194 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-res 4546 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 df-recs 6195 |
This theorem is referenced by: tfr0 6213 |
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