Metamath Proof Explorer


Theorem supsn

Description: The supremum of a singleton. (Contributed by NM, 2-Oct-2007)

Ref Expression
Assertion supsn ( ( 𝑅 Or 𝐴𝐵𝐴 ) → sup ( { 𝐵 } , 𝐴 , 𝑅 ) = 𝐵 )

Proof

Step Hyp Ref Expression
1 dfsn2 { 𝐵 } = { 𝐵 , 𝐵 }
2 1 supeq1i sup ( { 𝐵 } , 𝐴 , 𝑅 ) = sup ( { 𝐵 , 𝐵 } , 𝐴 , 𝑅 )
3 suppr ( ( 𝑅 Or 𝐴𝐵𝐴𝐵𝐴 ) → sup ( { 𝐵 , 𝐵 } , 𝐴 , 𝑅 ) = if ( 𝐵 𝑅 𝐵 , 𝐵 , 𝐵 ) )
4 3 3anidm23 ( ( 𝑅 Or 𝐴𝐵𝐴 ) → sup ( { 𝐵 , 𝐵 } , 𝐴 , 𝑅 ) = if ( 𝐵 𝑅 𝐵 , 𝐵 , 𝐵 ) )
5 2 4 eqtrid ( ( 𝑅 Or 𝐴𝐵𝐴 ) → sup ( { 𝐵 } , 𝐴 , 𝑅 ) = if ( 𝐵 𝑅 𝐵 , 𝐵 , 𝐵 ) )
6 ifid if ( 𝐵 𝑅 𝐵 , 𝐵 , 𝐵 ) = 𝐵
7 5 6 eqtrdi ( ( 𝑅 Or 𝐴𝐵𝐴 ) → sup ( { 𝐵 } , 𝐴 , 𝑅 ) = 𝐵 )