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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 4055 | . . . . 5 | |
2 | opexg 4150 | . . . . 5 | |
3 | 1, 2 | mpan 420 | . . . 4 |
4 | snidg 3554 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | fnsng 5170 | . . . . 5 | |
7 | 1, 6 | mpan 420 | . . . 4 |
8 | fvsng 5616 | . . . . . . 7 | |
9 | 1, 8 | mpan 420 | . . . . . 6 |
10 | res0 4823 | . . . . . . 7 | |
11 | 10 | fveq2i 5424 | . . . . . 6 |
12 | 9, 11 | syl6eqr 2190 | . . . . 5 |
13 | fveq2 5421 | . . . . . . 7 | |
14 | reseq2 4814 | . . . . . . . 8 | |
15 | 14 | fveq2d 5425 | . . . . . . 7 |
16 | 13, 15 | eqeq12d 2154 | . . . . . 6 |
17 | 1, 16 | ralsn 3567 | . . . . 5 |
18 | 12, 17 | sylibr 133 | . . . 4 |
19 | suc0 4333 | . . . . . 6 | |
20 | 0elon 4314 | . . . . . . 7 | |
21 | 20 | onsuci 4432 | . . . . . 6 |
22 | 19, 21 | eqeltrri 2213 | . . . . 5 |
23 | fneq2 5212 | . . . . . . 7 | |
24 | raleq 2626 | . . . . . . 7 | |
25 | 23, 24 | anbi12d 464 | . . . . . 6 |
26 | 25 | rspcev 2789 | . . . . 5 |
27 | 22, 26 | mpan 420 | . . . 4 |
28 | 7, 18, 27 | syl2anc 408 | . . 3 |
29 | snexg 4108 | . . . . 5 | |
30 | eleq2 2203 | . . . . . . 7 | |
31 | fneq1 5211 | . . . . . . . . 9 | |
32 | fveq1 5420 | . . . . . . . . . . 11 | |
33 | reseq1 4813 | . . . . . . . . . . . 12 | |
34 | 33 | fveq2d 5425 | . . . . . . . . . . 11 |
35 | 32, 34 | eqeq12d 2154 | . . . . . . . . . 10 |
36 | 35 | ralbidv 2437 | . . . . . . . . 9 |
37 | 31, 36 | anbi12d 464 | . . . . . . . 8 |
38 | 37 | rexbidv 2438 | . . . . . . 7 |
39 | 30, 38 | anbi12d 464 | . . . . . 6 |
40 | 39 | spcegv 2774 | . . . . 5 |
41 | 3, 29, 40 | 3syl 17 | . . . 4 |
42 | tfr.1 | . . . . . 6 recs | |
43 | 42 | eleq2i 2206 | . . . . 5 recs |
44 | df-recs 6202 | . . . . . 6 recs | |
45 | 44 | eleq2i 2206 | . . . . 5 recs |
46 | eluniab 3748 | . . . . 5 | |
47 | 43, 45, 46 | 3bitri 205 | . . . 4 |
48 | 41, 47 | syl6ibr 161 | . . 3 |
49 | 5, 28, 48 | mp2and 429 | . 2 |
50 | opeldmg 4744 | . . 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 2125 wral 2416 wrex 2417 cvv 2686 c0 3363 csn 3527 cop 3530 cuni 3736 con0 4285 csuc 4287 cdm 4539 cres 4541 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-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
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-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 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-br 3930 df-opab 3990 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-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-recs 6202 |
This theorem is referenced by: tfr0 6220 |
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