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Theorem smoel 6614
 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.)
Assertion
Ref Expression
smoel

Proof of Theorem smoel
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smodm 6605 . . . . 5
2 ordtr1 4616 . . . . . . 7
32ancomsd 441 . . . . . 6
43expdimp 427 . . . . 5
51, 4sylan 458 . . . 4
6 df-smo 6600 . . . . . 6
7 eleq1 2495 . . . . . . . . . . 11
8 fveq2 5720 . . . . . . . . . . . 12
98eleq1d 2501 . . . . . . . . . . 11
107, 9imbi12d 312 . . . . . . . . . 10
11 eleq2 2496 . . . . . . . . . . 11
12 fveq2 5720 . . . . . . . . . . . 12
1312eleq2d 2502 . . . . . . . . . . 11
1411, 13imbi12d 312 . . . . . . . . . 10
1510, 14rspc2v 3050 . . . . . . . . 9
1615ancoms 440 . . . . . . . 8
1716com12 29 . . . . . . 7
18173ad2ant3 980 . . . . . 6
196, 18sylbi 188 . . . . 5
2019expdimp 427 . . . 4
215, 20syld 42 . . 3
2221pm2.43d 46 . 2
23223impia 1150 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697   word 4572  con0 4573   cdm 4870  wf 5442  cfv 5446   wsmo 6599 This theorem is referenced by:  smoiun  6615  smoel2  6617 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-tr 4295  df-ord 4576  df-iota 5410  df-fv 5454  df-smo 6600
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