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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 6208 | . . 3 recs |
3 | ordsson 4403 | . . 3 recs recs | |
4 | 2, 3 | mp1i 10 | . 2 recs |
5 | tfrlemi14d.2 | . . . . . . . 8 | |
6 | 1, 5 | tfrlemi1 6222 | . . . . . . 7 |
7 | 5 | ad2antrr 479 | . . . . . . . . 9 |
8 | simplr 519 | . . . . . . . . 9 | |
9 | simprl 520 | . . . . . . . . 9 | |
10 | fneq2 5207 | . . . . . . . . . . . . 13 | |
11 | raleq 2624 | . . . . . . . . . . . . 13 | |
12 | 10, 11 | anbi12d 464 | . . . . . . . . . . . 12 |
13 | 12 | rspcev 2784 | . . . . . . . . . . 11 |
14 | 13 | adantll 467 | . . . . . . . . . 10 |
15 | vex 2684 | . . . . . . . . . . 11 | |
16 | 1, 15 | tfrlem3a 6200 | . . . . . . . . . 10 |
17 | 14, 16 | sylibr 133 | . . . . . . . . 9 |
18 | 1, 7, 8, 9, 17 | tfrlemisucaccv 6215 | . . . . . . . 8 |
19 | vex 2684 | . . . . . . . . . . . 12 | |
20 | 5 | tfrlem3-2d 6202 | . . . . . . . . . . . . 13 |
21 | 20 | simprd 113 | . . . . . . . . . . . 12 |
22 | opexg 4145 | . . . . . . . . . . . 12 | |
23 | 19, 21, 22 | sylancr 410 | . . . . . . . . . . 11 |
24 | snidg 3549 | . . . . . . . . . . 11 | |
25 | elun2 3239 | . . . . . . . . . . 11 | |
26 | 23, 24, 25 | 3syl 17 | . . . . . . . . . 10 |
27 | 26 | ad2antrr 479 | . . . . . . . . 9 |
28 | opeldmg 4739 | . . . . . . . . . . 11 | |
29 | 19, 21, 28 | sylancr 410 | . . . . . . . . . 10 |
30 | 29 | ad2antrr 479 | . . . . . . . . 9 |
31 | 27, 30 | mpd 13 | . . . . . . . 8 |
32 | dmeq 4734 | . . . . . . . . . 10 | |
33 | 32 | eleq2d 2207 | . . . . . . . . 9 |
34 | 33 | rspcev 2784 | . . . . . . . 8 |
35 | 18, 31, 34 | syl2anc 408 | . . . . . . 7 |
36 | 6, 35 | exlimddv 1870 | . . . . . 6 |
37 | eliun 3812 | . . . . . 6 | |
38 | 36, 37 | sylibr 133 | . . . . 5 |
39 | 38 | ex 114 | . . . 4 |
40 | 39 | ssrdv 3098 | . . 3 |
41 | 1 | recsfval 6205 | . . . . 5 recs |
42 | 41 | dmeqi 4735 | . . . 4 recs |
43 | dmuni 4744 | . . . 4 | |
44 | 42, 43 | eqtri 2158 | . . 3 recs |
45 | 40, 44 | sseqtrrdi 3141 | . 2 recs |
46 | 4, 45 | eqssd 3109 | 1 recs |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 wceq 1331 wcel 1480 cab 2123 wral 2414 wrex 2415 cvv 2681 cun 3064 wss 3066 csn 3522 cop 3525 cuni 3731 ciun 3808 word 4279 con0 4280 cdm 4534 cres 4536 wfun 5112 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-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
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-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 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-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 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-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-recs 6195 |
This theorem is referenced by: tfri1d 6225 |
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