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Theorem ex-un 25953
Description: Example for df-un 3395. 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 3582 . . 3  |-  ( ( { 1 ,  3 }  u.  { 1 } )  u.  {
8 } )  =  ( { 1 ,  3 }  u.  ( { 1 }  u.  { 8 } ) )
2 snsspr1 4112 . . . . 5  |-  { 1 }  C_  { 1 ,  3 }
3 ssequn2 3598 . . . . 5  |-  ( { 1 }  C_  { 1 ,  3 }  <->  ( {
1 ,  3 }  u.  { 1 } )  =  { 1 ,  3 } )
42, 3mpbi 213 . . . 4  |-  ( { 1 ,  3 }  u.  { 1 } )  =  { 1 ,  3 }
54uneq1i 3575 . . 3  |-  ( ( { 1 ,  3 }  u.  { 1 } )  u.  {
8 } )  =  ( { 1 ,  3 }  u.  {
8 } )
61, 5eqtr3i 2495 . 2  |-  ( { 1 ,  3 }  u.  ( { 1 }  u.  { 8 } ) )  =  ( { 1 ,  3 }  u.  {
8 } )
7 df-pr 3962 . . 3  |-  { 1 ,  8 }  =  ( { 1 }  u.  { 8 } )
87uneq2i 3576 . 2  |-  ( { 1 ,  3 }  u.  { 1 ,  8 } )  =  ( { 1 ,  3 }  u.  ( { 1 }  u.  { 8 } ) )
9 df-tp 3964 . 2  |-  { 1 ,  3 ,  8 }  =  ( { 1 ,  3 }  u.  { 8 } )
106, 8, 93eqtr4i 2503 1  |-  ( { 1 ,  3 }  u.  { 1 ,  8 } )  =  { 1 ,  3 ,  8 }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    u. cun 3388    C_ wss 3390   {csn 3959   {cpr 3961   {ctp 3963   1c1 9558   3c3 10682   8c8 10687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-un 3395  df-in 3397  df-ss 3404  df-pr 3962  df-tp 3964
This theorem is referenced by:  ex-uni  25955
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