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Mirrors > Home > MPE Home > Th. List > leiso | Structured version Visualization version Unicode version |
Description: Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
leiso |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-le 9681 |
. . . . . . 7
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2 | 1 | ineq1i 3630 |
. . . . . 6
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3 | indif1 3687 |
. . . . . 6
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4 | 2, 3 | eqtri 2473 |
. . . . 5
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5 | xpss12 4940 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | anidms 651 |
. . . . . . 7
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7 | dfss1 3637 |
. . . . . . 7
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8 | 6, 7 | sylib 200 |
. . . . . 6
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9 | 8 | difeq1d 3550 |
. . . . 5
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10 | 4, 9 | syl5req 2498 |
. . . 4
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11 | isoeq2 6211 |
. . . 4
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12 | 10, 11 | syl 17 |
. . 3
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13 | 1 | ineq1i 3630 |
. . . . . 6
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14 | indif1 3687 |
. . . . . 6
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15 | 13, 14 | eqtri 2473 |
. . . . 5
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16 | xpss12 4940 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16 | anidms 651 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | dfss1 3637 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | sylib 200 |
. . . . . 6
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20 | 19 | difeq1d 3550 |
. . . . 5
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21 | 15, 20 | syl5req 2498 |
. . . 4
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22 | isoeq3 6212 |
. . . 4
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23 | 21, 22 | syl 17 |
. . 3
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24 | 12, 23 | sylan9bb 706 |
. 2
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25 | isocnv2 6222 |
. . 3
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26 | eqid 2451 |
. . . 4
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27 | eqid 2451 |
. . . 4
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28 | 26, 27 | isocnv3 6223 |
. . 3
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29 | 25, 28 | bitri 253 |
. 2
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30 | isores1 6225 |
. . 3
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31 | isores2 6224 |
. . 3
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32 | 30, 31 | bitri 253 |
. 2
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33 | 24, 29, 32 | 3bitr4g 292 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-sbc 3268 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-isom 5591 df-le 9681 |
This theorem is referenced by: leisorel 12623 icopnfhmeo 21971 iccpnfhmeo 21973 xrhmeo 21974 |
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