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Theorem measle0 28416
Description: If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017.)
Assertion
Ref Expression
measle0  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  ( M `  A )  =  0 )

Proof of Theorem measle0
StepHypRef Expression
1 simp3 996 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  ( M `  A )  <_  0
)
2 measvxrge0 28413 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  ( M `  A )  e.  ( 0 [,] +oo ) )
3 elxrge0 11632 . . . . 5  |-  ( ( M `  A )  e.  ( 0 [,] +oo )  <->  ( ( M `
 A )  e. 
RR*  /\  0  <_  ( M `  A ) ) )
42, 3sylib 196 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
( M `  A
)  e.  RR*  /\  0  <_  ( M `  A
) ) )
543adant3 1014 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  ( ( M `  A )  e.  RR*  /\  0  <_ 
( M `  A
) ) )
65simprd 461 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  0  <_  ( M `  A ) )
75simpld 457 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  ( M `  A )  e.  RR* )
8 0xr 9629 . . 3  |-  0  e.  RR*
9 xrletri3 11361 . . 3  |-  ( ( ( M `  A
)  e.  RR*  /\  0  e.  RR* )  ->  (
( M `  A
)  =  0  <->  (
( M `  A
)  <_  0  /\  0  <_  ( M `  A ) ) ) )
107, 8, 9sylancl 660 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  ( ( M `  A )  =  0  <->  ( ( M `  A )  <_  0  /\  0  <_ 
( M `  A
) ) ) )
111, 6, 10mpbir2and 920 1  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  ( M `  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   0cc0 9481   +oocpnf 9614   RR*cxr 9616    <_ cle 9618   [,]cicc 11535  measurescmeas 28403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-icc 11539  df-esum 28257  df-meas 28404
This theorem is referenced by:  aean  28453
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