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Mirrors > Home > ILE Home > Th. List > smoel | Unicode version |
Description: If is less than then a strictly monotone function's value will be strictly less at than at . (Contributed by Andrew Salmon, 22-Nov-2011.) |
Ref | Expression |
---|---|
smoel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smodm 5906 | . . . . 5 | |
2 | ordtr1 4125 | . . . . . . 7 | |
3 | 2 | ancomsd 256 | . . . . . 6 |
4 | 3 | expdimp 246 | . . . . 5 |
5 | 1, 4 | sylan 267 | . . . 4 |
6 | df-smo 5901 | . . . . . 6 | |
7 | eleq1 2100 | . . . . . . . . . . 11 | |
8 | fveq2 5178 | . . . . . . . . . . . 12 | |
9 | 8 | eleq1d 2106 | . . . . . . . . . . 11 |
10 | 7, 9 | imbi12d 223 | . . . . . . . . . 10 |
11 | eleq2 2101 | . . . . . . . . . . 11 | |
12 | fveq2 5178 | . . . . . . . . . . . 12 | |
13 | 12 | eleq2d 2107 | . . . . . . . . . . 11 |
14 | 11, 13 | imbi12d 223 | . . . . . . . . . 10 |
15 | 10, 14 | rspc2v 2662 | . . . . . . . . 9 |
16 | 15 | ancoms 255 | . . . . . . . 8 |
17 | 16 | com12 27 | . . . . . . 7 |
18 | 17 | 3ad2ant3 927 | . . . . . 6 |
19 | 6, 18 | sylbi 114 | . . . . 5 |
20 | 19 | expdimp 246 | . . . 4 |
21 | 5, 20 | syld 40 | . . 3 |
22 | 21 | pm2.43d 44 | . 2 |
23 | 22 | 3impia 1101 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 wral 2306 word 4099 con0 4100 cdm 4345 wf 4898 cfv 4902 wsmo 5900 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-tr 3855 df-iord 4103 df-iota 4867 df-fv 4910 df-smo 5901 |
This theorem is referenced by: smoiun 5916 smoel2 5918 |
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