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Metamath Proof Explorer

Theorem volss

Description: The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017)

Ref Expression
Assertion volss 
|- ( ( A e. dom vol /\ B e. dom vol /\ A C_ B ) -> ( vol ` A ) <_ ( vol ` B ) )

Proof

Step Hyp Ref Expression
1 simp3 
 |-  ( ( A e. dom vol /\ B e. dom vol /\ A C_ B ) -> A C_ B )
2 mblss 
 |-  ( B e. dom vol -> B C_ RR )
3 2 3ad2ant2 
 |-  ( ( A e. dom vol /\ B e. dom vol /\ A C_ B ) -> B C_ RR )
4 ovolss 
 |-  ( ( A C_ B /\ B C_ RR ) -> ( vol* ` A ) <_ ( vol* ` B ) )
5 1 3 4 syl2anc 
 |-  ( ( A e. dom vol /\ B e. dom vol /\ A C_ B ) -> ( vol* ` A ) <_ ( vol* ` B ) )
6 mblvol 
 |-  ( A e. dom vol -> ( vol ` A ) = ( vol* ` A ) )
7 6 3ad2ant1 
 |-  ( ( A e. dom vol /\ B e. dom vol /\ A C_ B ) -> ( vol ` A ) = ( vol* ` A ) )
8 mblvol 
 |-  ( B e. dom vol -> ( vol ` B ) = ( vol* ` B ) )
9 8 3ad2ant2 
 |-  ( ( A e. dom vol /\ B e. dom vol /\ A C_ B ) -> ( vol ` B ) = ( vol* ` B ) )
10 5 7 9 3brtr4d 
 |-  ( ( A e. dom vol /\ B e. dom vol /\ A C_ B ) -> ( vol ` A ) <_ ( vol ` B ) )