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Mirrors > Home > ILE Home > Th. List > smores2 | Unicode version |
Description: A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) |
Ref | Expression |
---|---|
smores2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsmo2 6312 |
. . . . . . 7
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2 | 1 | simp1bi 1014 |
. . . . . 6
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3 | ffun 5387 |
. . . . . 6
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4 | 2, 3 | syl 14 |
. . . . 5
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5 | funres 5276 |
. . . . . 6
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6 | funfn 5265 |
. . . . . 6
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7 | 5, 6 | sylib 122 |
. . . . 5
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8 | 4, 7 | syl 14 |
. . . 4
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9 | df-ima 4657 |
. . . . . 6
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10 | imassrn 4999 |
. . . . . 6
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11 | 9, 10 | eqsstrri 3203 |
. . . . 5
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12 | frn 5393 |
. . . . . 6
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13 | 2, 12 | syl 14 |
. . . . 5
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14 | 11, 13 | sstrid 3181 |
. . . 4
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15 | df-f 5239 |
. . . 4
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16 | 8, 14, 15 | sylanbrc 417 |
. . 3
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17 | 16 | adantr 276 |
. 2
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18 | smodm 6316 |
. . 3
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19 | ordin 4403 |
. . . . 5
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20 | dmres 4946 |
. . . . . 6
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21 | ordeq 4390 |
. . . . . 6
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22 | 20, 21 | ax-mp 5 |
. . . . 5
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23 | 19, 22 | sylibr 134 |
. . . 4
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24 | 23 | ancoms 268 |
. . 3
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25 | 18, 24 | sylan 283 |
. 2
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26 | resss 4949 |
. . . . . 6
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27 | dmss 4844 |
. . . . . 6
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28 | 26, 27 | ax-mp 5 |
. . . . 5
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29 | 1 | simp3bi 1016 |
. . . . 5
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30 | ssralv 3234 |
. . . . 5
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31 | 28, 29, 30 | mpsyl 65 |
. . . 4
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32 | 31 | adantr 276 |
. . 3
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33 | ordtr1 4406 |
. . . . . . . . . . 11
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34 | 25, 33 | syl 14 |
. . . . . . . . . 10
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35 | inss1 3370 |
. . . . . . . . . . . 12
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36 | 20, 35 | eqsstri 3202 |
. . . . . . . . . . 11
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37 | 36 | sseli 3166 |
. . . . . . . . . 10
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38 | 34, 37 | syl6 33 |
. . . . . . . . 9
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39 | 38 | expcomd 1452 |
. . . . . . . 8
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40 | 39 | imp31 256 |
. . . . . . 7
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41 | fvres 5558 |
. . . . . . 7
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42 | 40, 41 | syl 14 |
. . . . . 6
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43 | 36 | sseli 3166 |
. . . . . . . 8
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44 | fvres 5558 |
. . . . . . . 8
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45 | 43, 44 | syl 14 |
. . . . . . 7
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46 | 45 | ad2antlr 489 |
. . . . . 6
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47 | 42, 46 | eleq12d 2260 |
. . . . 5
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48 | 47 | ralbidva 2486 |
. . . 4
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49 | 48 | ralbidva 2486 |
. . 3
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50 | 32, 49 | mpbird 167 |
. 2
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51 | dfsmo2 6312 |
. 2
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52 | 17, 25, 50, 51 | syl3anbrc 1183 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-tr 4117 df-iord 4384 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-smo 6311 |
This theorem is referenced by: (None) |
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