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Mirrors > Home > ILE Home > Th. List > tfrlemi14d | Unicode version |
Description: The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.) |
Ref | Expression |
---|---|
tfrlemi14d.1 | |
tfrlemi14d.2 |
Ref | Expression |
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tfrlemi14d | recs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlemi14d.1 | . . . 4 | |
2 | 1 | tfrlem8 5934 | . . 3 recs |
3 | ordsson 4218 | . . 3 recs recs | |
4 | 2, 3 | mp1i 10 | . 2 recs |
5 | tfrlemi14d.2 | . . . . . . . 8 | |
6 | 1, 5 | tfrlemi1 5946 | . . . . . . 7 |
7 | 5 | ad2antrr 457 | . . . . . . . . 9 |
8 | simplr 482 | . . . . . . . . 9 | |
9 | simprl 483 | . . . . . . . . 9 | |
10 | fneq2 4988 | . . . . . . . . . . . . 13 | |
11 | raleq 2505 | . . . . . . . . . . . . 13 | |
12 | 10, 11 | anbi12d 442 | . . . . . . . . . . . 12 |
13 | 12 | rspcev 2656 | . . . . . . . . . . 11 |
14 | 13 | adantll 445 | . . . . . . . . . 10 |
15 | vex 2560 | . . . . . . . . . . 11 | |
16 | 1, 15 | tfrlem3a 5925 | . . . . . . . . . 10 |
17 | 14, 16 | sylibr 137 | . . . . . . . . 9 |
18 | 1, 7, 8, 9, 17 | tfrlemisucaccv 5939 | . . . . . . . 8 |
19 | vex 2560 | . . . . . . . . . . . 12 | |
20 | 5 | tfrlem3-2d 5928 | . . . . . . . . . . . . 13 |
21 | 20 | simprd 107 | . . . . . . . . . . . 12 |
22 | opexg 3964 | . . . . . . . . . . . 12 | |
23 | 19, 21, 22 | sylancr 393 | . . . . . . . . . . 11 |
24 | snidg 3400 | . . . . . . . . . . 11 | |
25 | elun2 3111 | . . . . . . . . . . 11 | |
26 | 23, 24, 25 | 3syl 17 | . . . . . . . . . 10 |
27 | 26 | ad2antrr 457 | . . . . . . . . 9 |
28 | opeldmg 4540 | . . . . . . . . . . 11 | |
29 | 19, 21, 28 | sylancr 393 | . . . . . . . . . 10 |
30 | 29 | ad2antrr 457 | . . . . . . . . 9 |
31 | 27, 30 | mpd 13 | . . . . . . . 8 |
32 | dmeq 4535 | . . . . . . . . . 10 | |
33 | 32 | eleq2d 2107 | . . . . . . . . 9 |
34 | 33 | rspcev 2656 | . . . . . . . 8 |
35 | 18, 31, 34 | syl2anc 391 | . . . . . . 7 |
36 | 6, 35 | exlimddv 1778 | . . . . . 6 |
37 | eliun 3661 | . . . . . 6 | |
38 | 36, 37 | sylibr 137 | . . . . 5 |
39 | 38 | ex 108 | . . . 4 |
40 | 39 | ssrdv 2951 | . . 3 |
41 | 1 | recsfval 5931 | . . . . 5 recs |
42 | 41 | dmeqi 4536 | . . . 4 recs |
43 | dmuni 4545 | . . . 4 | |
44 | 42, 43 | eqtri 2060 | . . 3 recs |
45 | 40, 44 | syl6sseqr 2992 | . 2 recs |
46 | 4, 45 | eqssd 2962 | 1 recs |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wal 1241 wceq 1243 wcel 1393 cab 2026 wral 2306 wrex 2307 cvv 2557 cun 2915 wss 2917 csn 3375 cop 3378 cuni 3580 ciun 3657 word 4099 con0 4100 cdm 4345 cres 4347 wfun 4896 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-coll 3872 ax-sep 3875 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-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 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-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-recs 5920 |
This theorem is referenced by: tfri1d 5949 |
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