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Metamath Proof Explorer

Theorem ex-un

Description: Example for df-un . Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015)

Ref Expression
Assertion ex-un 
|- ( { 1 , 3 } u. { 1 , 8 } ) = { 1 , 3 , 8 }

Proof

Step Hyp Ref Expression
1 unass 
 |-  ( ( { 1 , 3 } u. { 1 } ) u. { 8 } ) = ( { 1 , 3 } u. ( { 1 } u. { 8 } ) )
2 snsspr1 
 |-  { 1 } C_ { 1 , 3 }
3 ssequn2 
 |-  ( { 1 } C_ { 1 , 3 } <-> ( { 1 , 3 } u. { 1 } ) = { 1 , 3 } )
4 2 3 mpbi 
 |-  ( { 1 , 3 } u. { 1 } ) = { 1 , 3 }
5 4 uneq1i 
 |-  ( ( { 1 , 3 } u. { 1 } ) u. { 8 } ) = ( { 1 , 3 } u. { 8 } )
6 1 5 eqtr3i 
 |-  ( { 1 , 3 } u. ( { 1 } u. { 8 } ) ) = ( { 1 , 3 } u. { 8 } )
7 df-pr 
 |-  { 1 , 8 } = ( { 1 } u. { 8 } )
8 7 uneq2i 
 |-  ( { 1 , 3 } u. { 1 , 8 } ) = ( { 1 , 3 } u. ( { 1 } u. { 8 } ) )
9 df-tp 
 |-  { 1 , 3 , 8 } = ( { 1 , 3 } u. { 8 } )
10 6 8 9 3eqtr4i 
 |-  ( { 1 , 3 } u. { 1 , 8 } ) = { 1 , 3 , 8 }