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Mirrors > Home > MPE Home > Th. List > smores2 | Structured version Visualization version 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 7071 |
. . . . . . 7
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2 | 1 | simp1bi 1024 |
. . . . . 6
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3 | ffun 5736 |
. . . . . 6
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4 | 2, 3 | syl 17 |
. . . . 5
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5 | funres 5624 |
. . . . . 6
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6 | funfn 5614 |
. . . . . 6
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7 | 5, 6 | sylib 200 |
. . . . 5
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8 | 4, 7 | syl 17 |
. . . 4
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9 | df-ima 4850 |
. . . . . 6
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10 | imassrn 5182 |
. . . . . 6
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11 | 9, 10 | eqsstr3i 3465 |
. . . . 5
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12 | frn 5740 |
. . . . . 6
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13 | 2, 12 | syl 17 |
. . . . 5
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14 | 11, 13 | syl5ss 3445 |
. . . 4
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15 | df-f 5589 |
. . . 4
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16 | 8, 14, 15 | sylanbrc 671 |
. . 3
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17 | 16 | adantr 467 |
. 2
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18 | smodm 7075 |
. . 3
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19 | ordin 5456 |
. . . . 5
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20 | dmres 5128 |
. . . . . 6
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21 | ordeq 5433 |
. . . . . 6
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22 | 20, 21 | ax-mp 5 |
. . . . 5
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23 | 19, 22 | sylibr 216 |
. . . 4
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24 | 23 | ancoms 455 |
. . 3
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25 | 18, 24 | sylan 474 |
. 2
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26 | resss 5131 |
. . . . . 6
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27 | dmss 5037 |
. . . . . 6
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28 | 26, 27 | ax-mp 5 |
. . . . 5
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29 | 1 | simp3bi 1026 |
. . . . 5
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30 | ssralv 3495 |
. . . . 5
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31 | 28, 29, 30 | mpsyl 65 |
. . . 4
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32 | 31 | adantr 467 |
. . 3
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33 | ordtr1 5469 |
. . . . . . . . . . 11
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34 | 25, 33 | syl 17 |
. . . . . . . . . 10
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35 | inss1 3654 |
. . . . . . . . . . . 12
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36 | 20, 35 | eqsstri 3464 |
. . . . . . . . . . 11
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37 | 36 | sseli 3430 |
. . . . . . . . . 10
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38 | 34, 37 | syl6 34 |
. . . . . . . . 9
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39 | 38 | expcomd 440 |
. . . . . . . 8
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40 | 39 | imp31 434 |
. . . . . . 7
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41 | fvres 5884 |
. . . . . . 7
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42 | 40, 41 | syl 17 |
. . . . . 6
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43 | 36 | sseli 3430 |
. . . . . . . 8
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44 | fvres 5884 |
. . . . . . . 8
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45 | 43, 44 | syl 17 |
. . . . . . 7
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46 | 45 | ad2antlr 734 |
. . . . . 6
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47 | 42, 46 | eleq12d 2525 |
. . . . 5
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48 | 47 | ralbidva 2826 |
. . . 4
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49 | 48 | ralbidva 2826 |
. . 3
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50 | 32, 49 | mpbird 236 |
. 2
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51 | dfsmo2 7071 |
. 2
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52 | 17, 25, 50, 51 | syl3anbrc 1193 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pr 4642 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-tr 4501 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-ord 5429 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-fv 5593 df-smo 7070 |
This theorem is referenced by: (None) |
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