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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 6156 | . . . . 5 | |
2 | ordtr1 4280 | . . . . . . 7 | |
3 | 2 | ancomsd 267 | . . . . . 6 |
4 | 3 | expdimp 257 | . . . . 5 |
5 | 1, 4 | sylan 281 | . . . 4 |
6 | df-smo 6151 | . . . . . 6 | |
7 | eleq1 2180 | . . . . . . . . . . 11 | |
8 | fveq2 5389 | . . . . . . . . . . . 12 | |
9 | 8 | eleq1d 2186 | . . . . . . . . . . 11 |
10 | 7, 9 | imbi12d 233 | . . . . . . . . . 10 |
11 | eleq2 2181 | . . . . . . . . . . 11 | |
12 | fveq2 5389 | . . . . . . . . . . . 12 | |
13 | 12 | eleq2d 2187 | . . . . . . . . . . 11 |
14 | 11, 13 | imbi12d 233 | . . . . . . . . . 10 |
15 | 10, 14 | rspc2v 2776 | . . . . . . . . 9 |
16 | 15 | ancoms 266 | . . . . . . . 8 |
17 | 16 | com12 30 | . . . . . . 7 |
18 | 17 | 3ad2ant3 989 | . . . . . 6 |
19 | 6, 18 | sylbi 120 | . . . . 5 |
20 | 19 | expdimp 257 | . . . 4 |
21 | 5, 20 | syld 45 | . . 3 |
22 | 21 | pm2.43d 50 | . 2 |
23 | 22 | 3impia 1163 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 wral 2393 word 4254 con0 4255 cdm 4509 wf 5089 cfv 5093 wsmo 6150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-tr 3997 df-iord 4258 df-iota 5058 df-fv 5101 df-smo 6151 |
This theorem is referenced by: smoiun 6166 smoel2 6168 |
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