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Theorem ex-un 20827
Description: Example for df-un 3170. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
ex-un  |-  ( { 1 ,  3 }  u.  { 1 ,  8 } )  =  { 1 ,  3 ,  8 }

Proof of Theorem ex-un
StepHypRef Expression
1 unass 3345 . . 3  |-  ( ( { 1 ,  3 }  u.  { 1 } )  u.  {
8 } )  =  ( { 1 ,  3 }  u.  ( { 1 }  u.  { 8 } ) )
2 snsspr1 3780 . . . . 5  |-  { 1 }  C_  { 1 ,  3 }
3 ssequn2 3361 . . . . 5  |-  ( { 1 }  C_  { 1 ,  3 }  <->  ( {
1 ,  3 }  u.  { 1 } )  =  { 1 ,  3 } )
42, 3mpbi 199 . . . 4  |-  ( { 1 ,  3 }  u.  { 1 } )  =  { 1 ,  3 }
54uneq1i 3338 . . 3  |-  ( ( { 1 ,  3 }  u.  { 1 } )  u.  {
8 } )  =  ( { 1 ,  3 }  u.  {
8 } )
61, 5eqtr3i 2318 . 2  |-  ( { 1 ,  3 }  u.  ( { 1 }  u.  { 8 } ) )  =  ( { 1 ,  3 }  u.  {
8 } )
7 df-pr 3660 . . 3  |-  { 1 ,  8 }  =  ( { 1 }  u.  { 8 } )
87uneq2i 3339 . 2  |-  ( { 1 ,  3 }  u.  { 1 ,  8 } )  =  ( { 1 ,  3 }  u.  ( { 1 }  u.  { 8 } ) )
9 df-tp 3661 . 2  |-  { 1 ,  3 ,  8 }  =  ( { 1 ,  3 }  u.  { 8 } )
106, 8, 93eqtr4i 2326 1  |-  ( { 1 ,  3 }  u.  { 1 ,  8 } )  =  { 1 ,  3 ,  8 }
Colors of variables: wff set class
Syntax hints:    = wceq 1632    u. cun 3163    C_ wss 3165   {csn 3653   {cpr 3654   {ctp 3655   1c1 8754   3c3 9812   8c8 9817
This theorem is referenced by:  ex-uni  20829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-pr 3660  df-tp 3661
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