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Mirrors > Home > ILE Home > Th. List > oawordi | Unicode version |
Description: Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.) |
Ref | Expression |
---|---|
oawordi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oafnex 6420 | . . . . 5 | |
2 | 1 | a1i 9 | . . . 4 |
3 | simpl3 997 | . . . 4 | |
4 | simpl1 995 | . . . 4 | |
5 | simpl2 996 | . . . 4 | |
6 | simpr 109 | . . . 4 | |
7 | 2, 3, 4, 5, 6 | rdgss 6359 | . . 3 |
8 | 3, 4 | jca 304 | . . . 4 |
9 | oav 6430 | . . . 4 | |
10 | 8, 9 | syl 14 | . . 3 |
11 | 3, 5 | jca 304 | . . . 4 |
12 | oav 6430 | . . . 4 | |
13 | 11, 12 | syl 14 | . . 3 |
14 | 7, 10, 13 | 3sstr4d 3192 | . 2 |
15 | 14 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 cvv 2730 wss 3121 cmpt 4048 con0 4346 csuc 4348 wfn 5191 cfv 5196 (class class class)co 5850 crdg 6345 coa 6389 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-recs 6281 df-irdg 6346 df-oadd 6396 |
This theorem is referenced by: oaword1 6447 |
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