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Mirrors > Home > ILE Home > Th. List > oeicl | Unicode version |
Description: Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.) |
Ref | Expression |
---|---|
oeicl | ↑o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oeiv 6352 | . 2 ↑o | |
2 | 1on 6320 | . . . 4 | |
3 | 2 | a1i 9 | . . 3 |
4 | vex 2689 | . . . . . . 7 | |
5 | omcl 6357 | . . . . . . 7 | |
6 | oveq1 5781 | . . . . . . . 8 | |
7 | eqid 2139 | . . . . . . . 8 | |
8 | 6, 7 | fvmptg 5497 | . . . . . . 7 |
9 | 4, 5, 8 | sylancr 410 | . . . . . 6 |
10 | 9, 5 | eqeltrd 2216 | . . . . 5 |
11 | 10 | ancoms 266 | . . . 4 |
12 | 11 | ralrimiva 2505 | . . 3 |
13 | 3, 12 | rdgon 6283 | . 2 |
14 | 1, 13 | eqeltrd 2216 | 1 ↑o |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 cmpt 3989 con0 4285 cfv 5123 (class class class)co 5774 crdg 6266 c1o 6306 comu 6311 ↑o coei 6312 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 df-oexpi 6319 |
This theorem is referenced by: (None) |
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