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Mirrors > Home > MPE Home > Th. List > oewordi | Structured version Unicode version |
Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) |
Ref | Expression |
---|---|
oewordi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4827 |
. . . . . 6
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2 | ordgt0ge1 7037 |
. . . . . 6
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3 | 1, 2 | syl 16 |
. . . . 5
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4 | 1on 7027 |
. . . . . 6
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5 | onsseleq 4858 |
. . . . . 6
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6 | 4, 5 | mpan 670 |
. . . . 5
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7 | 3, 6 | bitrd 253 |
. . . 4
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8 | 7 | 3ad2ant3 1011 |
. . 3
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9 | ondif2 7042 |
. . . . . . 7
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10 | oeword 7129 |
. . . . . . . . 9
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11 | 10 | biimpd 207 |
. . . . . . . 8
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12 | 11 | 3expia 1190 |
. . . . . . 7
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13 | 9, 12 | syl5bir 218 |
. . . . . 6
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14 | 13 | expd 436 |
. . . . 5
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15 | 14 | 3impia 1185 |
. . . 4
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16 | oe1m 7084 |
. . . . . . . . . 10
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17 | 16 | adantr 465 |
. . . . . . . . 9
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18 | oe1m 7084 |
. . . . . . . . . 10
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19 | 18 | adantl 466 |
. . . . . . . . 9
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20 | 17, 19 | eqtr4d 2495 |
. . . . . . . 8
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21 | eqimss 3506 |
. . . . . . . 8
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22 | 20, 21 | syl 16 |
. . . . . . 7
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23 | oveq1 6197 |
. . . . . . . 8
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24 | oveq1 6197 |
. . . . . . . 8
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25 | 23, 24 | sseq12d 3483 |
. . . . . . 7
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26 | 22, 25 | syl5ibcom 220 |
. . . . . 6
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27 | 26 | 3adant3 1008 |
. . . . 5
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28 | 27 | a1dd 46 |
. . . 4
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29 | 15, 28 | jaod 380 |
. . 3
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30 | 8, 29 | sylbid 215 |
. 2
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31 | 30 | imp 429 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4501 ax-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3070 df-sbc 3285 df-csb 3387 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-pss 3442 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-tp 3980 df-op 3982 df-uni 4190 df-iun 4271 df-br 4391 df-opab 4449 df-mpt 4450 df-tr 4484 df-eprel 4730 df-id 4734 df-po 4739 df-so 4740 df-fr 4777 df-we 4779 df-ord 4820 df-on 4821 df-lim 4822 df-suc 4823 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-rn 4949 df-res 4950 df-ima 4951 df-iota 5479 df-fun 5518 df-fn 5519 df-f 5520 df-f1 5521 df-fo 5522 df-f1o 5523 df-fv 5524 df-ov 6193 df-oprab 6194 df-mpt2 6195 df-om 6577 df-recs 6932 df-rdg 6966 df-1o 7020 df-2o 7021 df-oadd 7024 df-omul 7025 df-oexp 7026 |
This theorem is referenced by: oelim2 7134 oeoalem 7135 oeoelem 7137 oaabs2 7184 cantnflt 7981 cantnfltOLD 8011 cnfcom 8034 cnfcomOLD 8042 |
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