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Theorem measle0 26760
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 990 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  ( M `  A )  <_  0
)
2 measvxrge0 26757 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  ( M `  A )  e.  ( 0 [,] +oo ) )
3 elxrge0 11504 . . . . 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 1008 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  ( ( M `  A )  e.  RR*  /\  0  <_ 
( M `  A
) ) )
65simprd 463 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  0  <_  ( M `  A ) )
75simpld 459 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  ( M `  A )  e.  RR* )
8 0xr 9534 . . 3  |-  0  e.  RR*
9 xrletri3 11233 . . 3  |-  ( ( ( M `  A
)  e.  RR*  /\  0  e.  RR* )  ->  (
( M `  A
)  =  0  <->  (
( M `  A
)  <_  0  /\  0  <_  ( M `  A ) ) ) )
107, 8, 9sylancl 662 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  <_  0
)  ->  ( ( M `  A )  =  0  <->  ( ( M `  A )  <_  0  /\  0  <_ 
( M `  A
) ) ) )
111, 6, 10mpbir2and 913 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   0cc0 9386   +oocpnf 9519   RR*cxr 9521    <_ cle 9523   [,]cicc 11407  measurescmeas 26747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-i2m1 9454  ax-1ne0 9455  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-icc 11411  df-esum 26622  df-meas 26748
This theorem is referenced by:  aean  26797
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