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Theorem ex-un 20811
Description: Example for df-un 3157. 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 3332 . . 3  |-  ( ( { 1 ,  3 }  u.  { 1 } )  u.  {
8 } )  =  ( { 1 ,  3 }  u.  ( { 1 }  u.  { 8 } ) )
2 snsspr1 3764 . . . . 5  |-  { 1 }  C_  { 1 ,  3 }
3 ssequn2 3348 . . . . 5  |-  ( { 1 }  C_  { 1 ,  3 }  <->  ( {
1 ,  3 }  u.  { 1 } )  =  { 1 ,  3 } )
42, 3mpbi 199 . . . 4  |-  ( { 1 ,  3 }  u.  { 1 } )  =  { 1 ,  3 }
54uneq1i 3325 . . 3  |-  ( ( { 1 ,  3 }  u.  { 1 } )  u.  {
8 } )  =  ( { 1 ,  3 }  u.  {
8 } )
61, 5eqtr3i 2305 . 2  |-  ( { 1 ,  3 }  u.  ( { 1 }  u.  { 8 } ) )  =  ( { 1 ,  3 }  u.  {
8 } )
7 df-pr 3647 . . 3  |-  { 1 ,  8 }  =  ( { 1 }  u.  { 8 } )
87uneq2i 3326 . 2  |-  ( { 1 ,  3 }  u.  { 1 ,  8 } )  =  ( { 1 ,  3 }  u.  ( { 1 }  u.  { 8 } ) )
9 df-tp 3648 . 2  |-  { 1 ,  3 ,  8 }  =  ( { 1 ,  3 }  u.  { 8 } )
106, 8, 93eqtr4i 2313 1  |-  ( { 1 ,  3 }  u.  { 1 ,  8 } )  =  { 1 ,  3 ,  8 }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    u. cun 3150    C_ wss 3152   {csn 3640   {cpr 3641   {ctp 3642   1c1 8738   3c3 9796   8c8 9801
This theorem is referenced by:  ex-uni  20813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-pr 3647  df-tp 3648
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