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Mirrors > Home > ILE Home > Th. List > tfr0 | Unicode version |
Description: Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.) |
Ref | Expression |
---|---|
tfr.1 | recs |
Ref | Expression |
---|---|
tfr0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 3884 | . . . . 5 | |
2 | opexg 3964 | . . . . 5 | |
3 | 1, 2 | mpan 400 | . . . 4 |
4 | snidg 3400 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | fnsng 4947 | . . . . 5 | |
7 | 1, 6 | mpan 400 | . . . 4 |
8 | fvsng 5359 | . . . . . . 7 | |
9 | 1, 8 | mpan 400 | . . . . . 6 |
10 | res0 4616 | . . . . . . 7 | |
11 | 10 | fveq2i 5181 | . . . . . 6 |
12 | 9, 11 | syl6eqr 2090 | . . . . 5 |
13 | fveq2 5178 | . . . . . . 7 | |
14 | reseq2 4607 | . . . . . . . 8 | |
15 | 14 | fveq2d 5182 | . . . . . . 7 |
16 | 13, 15 | eqeq12d 2054 | . . . . . 6 |
17 | 1, 16 | ralsn 3414 | . . . . 5 |
18 | 12, 17 | sylibr 137 | . . . 4 |
19 | suc0 4148 | . . . . . 6 | |
20 | 0elon 4129 | . . . . . . 7 | |
21 | 20 | onsuci 4242 | . . . . . 6 |
22 | 19, 21 | eqeltrri 2111 | . . . . 5 |
23 | fneq2 4988 | . . . . . . 7 | |
24 | raleq 2505 | . . . . . . 7 | |
25 | 23, 24 | anbi12d 442 | . . . . . 6 |
26 | 25 | rspcev 2656 | . . . . 5 |
27 | 22, 26 | mpan 400 | . . . 4 |
28 | 7, 18, 27 | syl2anc 391 | . . 3 |
29 | snexg 3936 | . . . . 5 | |
30 | eleq2 2101 | . . . . . . 7 | |
31 | fneq1 4987 | . . . . . . . . 9 | |
32 | fveq1 5177 | . . . . . . . . . . 11 | |
33 | reseq1 4606 | . . . . . . . . . . . 12 | |
34 | 33 | fveq2d 5182 | . . . . . . . . . . 11 |
35 | 32, 34 | eqeq12d 2054 | . . . . . . . . . 10 |
36 | 35 | ralbidv 2326 | . . . . . . . . 9 |
37 | 31, 36 | anbi12d 442 | . . . . . . . 8 |
38 | 37 | rexbidv 2327 | . . . . . . 7 |
39 | 30, 38 | anbi12d 442 | . . . . . 6 |
40 | 39 | spcegv 2641 | . . . . 5 |
41 | 3, 29, 40 | 3syl 17 | . . . 4 |
42 | tfr.1 | . . . . . 6 recs | |
43 | 42 | eleq2i 2104 | . . . . 5 recs |
44 | df-recs 5920 | . . . . . 6 recs | |
45 | 44 | eleq2i 2104 | . . . . 5 recs |
46 | eluniab 3592 | . . . . 5 | |
47 | 43, 45, 46 | 3bitri 195 | . . . 4 |
48 | 41, 47 | syl6ibr 151 | . . 3 |
49 | 5, 28, 48 | mp2and 409 | . 2 |
50 | opeldmg 4540 | . . . . 5 | |
51 | 1, 50 | mpan 400 | . . . 4 |
52 | 42 | tfr2a 5936 | . . . 4 |
53 | 51, 52 | syl6 29 | . . 3 |
54 | res0 4616 | . . . . 5 | |
55 | 54 | fveq2i 5181 | . . . 4 |
56 | 55 | eqeq2i 2050 | . . 3 |
57 | 53, 56 | syl6ib 150 | . 2 |
58 | 49, 57 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 cab 2026 wral 2306 wrex 2307 cvv 2557 c0 3224 csn 3375 cop 3378 cuni 3580 con0 4100 csuc 4102 cdm 4345 cres 4347 wfn 4897 cfv 4902 recscrecs 5919 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 df-recs 5920 |
This theorem is referenced by: rdg0 5974 frec0g 5983 |
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