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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 5843 |
. . . . . . 7
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2 | 1 | simp1bi 918 |
. . . . . 6
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3 | ffun 4991 |
. . . . . 6
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4 | 2, 3 | syl 14 |
. . . . 5
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5 | funres 4884 |
. . . . . 6
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6 | funfn 4874 |
. . . . . 6
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7 | 5, 6 | sylib 127 |
. . . . 5
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8 | 4, 7 | syl 14 |
. . . 4
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9 | df-ima 4301 |
. . . . . 6
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10 | imassrn 4622 |
. . . . . 6
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11 | 9, 10 | eqsstr3i 2970 |
. . . . 5
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12 | frn 4995 |
. . . . . 6
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13 | 2, 12 | syl 14 |
. . . . 5
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14 | 11, 13 | syl5ss 2950 |
. . . 4
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15 | df-f 4849 |
. . . 4
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16 | 8, 14, 15 | sylanbrc 394 |
. . 3
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17 | 16 | adantr 261 |
. 2
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18 | smodm 5847 |
. . 3
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19 | ordin 4088 |
. . . . 5
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20 | dmres 4575 |
. . . . . 6
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21 | ordeq 4075 |
. . . . . 6
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22 | 20, 21 | ax-mp 7 |
. . . . 5
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23 | 19, 22 | sylibr 137 |
. . . 4
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24 | 23 | ancoms 255 |
. . 3
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25 | 18, 24 | sylan 267 |
. 2
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26 | resss 4578 |
. . . . . 6
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27 | dmss 4477 |
. . . . . 6
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28 | 26, 27 | ax-mp 7 |
. . . . 5
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29 | 1 | simp3bi 920 |
. . . . 5
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30 | ssralv 2998 |
. . . . 5
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31 | 28, 29, 30 | mpsyl 59 |
. . . 4
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32 | 31 | adantr 261 |
. . 3
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33 | ordtr1 4091 |
. . . . . . . . . . 11
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34 | 25, 33 | syl 14 |
. . . . . . . . . 10
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35 | inss1 3151 |
. . . . . . . . . . . 12
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36 | 20, 35 | eqsstri 2969 |
. . . . . . . . . . 11
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37 | 36 | sseli 2935 |
. . . . . . . . . 10
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38 | 34, 37 | syl6 29 |
. . . . . . . . 9
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39 | 38 | expcomd 1327 |
. . . . . . . 8
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40 | 39 | imp31 243 |
. . . . . . 7
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41 | fvres 5141 |
. . . . . . 7
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42 | 40, 41 | syl 14 |
. . . . . 6
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43 | 36 | sseli 2935 |
. . . . . . . 8
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44 | fvres 5141 |
. . . . . . . 8
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45 | 43, 44 | syl 14 |
. . . . . . 7
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46 | 45 | ad2antlr 458 |
. . . . . 6
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47 | 42, 46 | eleq12d 2105 |
. . . . 5
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48 | 47 | ralbidva 2316 |
. . . 4
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49 | 48 | ralbidva 2316 |
. . 3
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50 | 32, 49 | mpbird 156 |
. 2
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51 | dfsmo2 5843 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
52 | 17, 25, 50, 51 | syl3anbrc 1087 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-tr 3846 df-iord 4069 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 df-smo 5842 |
This theorem is referenced by: (None) |
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