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Mirrors > Home > ILE Home > Th. List > tfrlem8 | Unicode version |
Description: Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.) |
Ref | Expression |
---|---|
tfrlem.1 |
Ref | Expression |
---|---|
tfrlem8 | recs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem.1 | . . . . . . . . 9 | |
2 | 1 | tfrlem3 5926 | . . . . . . . 8 |
3 | 2 | abeq2i 2148 | . . . . . . 7 |
4 | fndm 4998 | . . . . . . . . . . 11 | |
5 | 4 | adantr 261 | . . . . . . . . . 10 |
6 | 5 | eleq1d 2106 | . . . . . . . . 9 |
7 | 6 | biimprcd 149 | . . . . . . . 8 |
8 | 7 | rexlimiv 2427 | . . . . . . 7 |
9 | 3, 8 | sylbi 114 | . . . . . 6 |
10 | eleq1a 2109 | . . . . . 6 | |
11 | 9, 10 | syl 14 | . . . . 5 |
12 | 11 | rexlimiv 2427 | . . . 4 |
13 | 12 | abssi 3015 | . . 3 |
14 | ssorduni 4213 | . . 3 | |
15 | 13, 14 | ax-mp 7 | . 2 |
16 | 1 | recsfval 5931 | . . . . 5 recs |
17 | 16 | dmeqi 4536 | . . . 4 recs |
18 | dmuni 4545 | . . . 4 | |
19 | vex 2560 | . . . . . 6 | |
20 | 19 | dmex 4598 | . . . . 5 |
21 | 20 | dfiun2 3691 | . . . 4 |
22 | 17, 18, 21 | 3eqtri 2064 | . . 3 recs |
23 | ordeq 4109 | . . 3 recs recs | |
24 | 22, 23 | ax-mp 7 | . 2 recs |
25 | 15, 24 | mpbir 134 | 1 recs |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 cab 2026 wral 2306 wrex 2307 wss 2917 cuni 3580 ciun 3657 word 4099 con0 4100 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-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-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 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-v 2559 df-un 2922 df-in 2924 df-ss 2931 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-tr 3855 df-iord 4103 df-on 4105 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 df-recs 5920 |
This theorem is referenced by: tfrlemi14d 5947 |
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