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Mirrors > Home > ILE Home > Th. List > oav2 | Unicode version |
Description: Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.) |
Ref | Expression |
---|---|
oav2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oafnex 6340 | . . 3 | |
2 | rdgival 6279 | . . 3 | |
3 | 1, 2 | mp3an1 1302 | . 2 |
4 | oav 6350 | . 2 | |
5 | onelon 4306 | . . . . . 6 | |
6 | vex 2689 | . . . . . . . . . 10 | |
7 | oaexg 6344 | . . . . . . . . . 10 | |
8 | 6, 7 | mpan2 421 | . . . . . . . . 9 |
9 | sucexg 4414 | . . . . . . . . . 10 | |
10 | 8, 9 | syl 14 | . . . . . . . . 9 |
11 | suceq 4324 | . . . . . . . . . 10 | |
12 | eqid 2139 | . . . . . . . . . 10 | |
13 | 11, 12 | fvmptg 5497 | . . . . . . . . 9 |
14 | 8, 10, 13 | syl2anc 408 | . . . . . . . 8 |
15 | 14 | adantr 274 | . . . . . . 7 |
16 | oav 6350 | . . . . . . . 8 | |
17 | 16 | fveq2d 5425 | . . . . . . 7 |
18 | 15, 17 | eqtr3d 2174 | . . . . . 6 |
19 | 5, 18 | sylan2 284 | . . . . 5 |
20 | 19 | anassrs 397 | . . . 4 |
21 | 20 | iuneq2dv 3834 | . . 3 |
22 | 21 | uneq2d 3230 | . 2 |
23 | 3, 4, 22 | 3eqtr4d 2182 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 cun 3069 ciun 3813 cmpt 3989 con0 4285 csuc 4287 wfn 5118 cfv 5123 (class class class)co 5774 crdg 6266 coa 6310 |
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-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-oadd 6317 |
This theorem is referenced by: oasuc 6360 |
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