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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 4103 | . . . . 5 | |
2 | opexg 4200 | . . . . 5 | |
3 | 1, 2 | mpan 421 | . . . 4 |
4 | snidg 3599 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | fnsng 5229 | . . . . 5 | |
7 | 1, 6 | mpan 421 | . . . 4 |
8 | fvsng 5675 | . . . . . . 7 | |
9 | 1, 8 | mpan 421 | . . . . . 6 |
10 | res0 4882 | . . . . . . 7 | |
11 | 10 | fveq2i 5483 | . . . . . 6 |
12 | 9, 11 | eqtr4di 2215 | . . . . 5 |
13 | fveq2 5480 | . . . . . . 7 | |
14 | reseq2 4873 | . . . . . . . 8 | |
15 | 14 | fveq2d 5484 | . . . . . . 7 |
16 | 13, 15 | eqeq12d 2179 | . . . . . 6 |
17 | 1, 16 | ralsn 3613 | . . . . 5 |
18 | 12, 17 | sylibr 133 | . . . 4 |
19 | suc0 4383 | . . . . . 6 | |
20 | 0elon 4364 | . . . . . . 7 | |
21 | 20 | onsuci 4487 | . . . . . 6 |
22 | 19, 21 | eqeltrri 2238 | . . . . 5 |
23 | fneq2 5271 | . . . . . . 7 | |
24 | raleq 2659 | . . . . . . 7 | |
25 | 23, 24 | anbi12d 465 | . . . . . 6 |
26 | 25 | rspcev 2825 | . . . . 5 |
27 | 22, 26 | mpan 421 | . . . 4 |
28 | 7, 18, 27 | syl2anc 409 | . . 3 |
29 | snexg 4157 | . . . . 5 | |
30 | eleq2 2228 | . . . . . . 7 | |
31 | fneq1 5270 | . . . . . . . . 9 | |
32 | fveq1 5479 | . . . . . . . . . . 11 | |
33 | reseq1 4872 | . . . . . . . . . . . 12 | |
34 | 33 | fveq2d 5484 | . . . . . . . . . . 11 |
35 | 32, 34 | eqeq12d 2179 | . . . . . . . . . 10 |
36 | 35 | ralbidv 2464 | . . . . . . . . 9 |
37 | 31, 36 | anbi12d 465 | . . . . . . . 8 |
38 | 37 | rexbidv 2465 | . . . . . . 7 |
39 | 30, 38 | anbi12d 465 | . . . . . 6 |
40 | 39 | spcegv 2809 | . . . . 5 |
41 | 3, 29, 40 | 3syl 17 | . . . 4 |
42 | tfr.1 | . . . . . 6 recs | |
43 | 42 | eleq2i 2231 | . . . . 5 recs |
44 | df-recs 6264 | . . . . . 6 recs | |
45 | 44 | eleq2i 2231 | . . . . 5 recs |
46 | eluniab 3795 | . . . . 5 | |
47 | 43, 45, 46 | 3bitri 205 | . . . 4 |
48 | 41, 47 | syl6ibr 161 | . . 3 |
49 | 5, 28, 48 | mp2and 430 | . 2 |
50 | opeldmg 4803 | . . 3 | |
51 | 1, 50 | mpan 421 | . 2 |
52 | 49, 51 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wex 1479 wcel 2135 cab 2150 wral 2442 wrex 2443 cvv 2721 c0 3404 csn 3570 cop 3573 cuni 3783 con0 4335 csuc 4337 cdm 4598 cres 4600 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-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
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-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 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-br 3977 df-opab 4038 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-res 4610 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 df-recs 6264 |
This theorem is referenced by: tfr0 6282 |
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