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Theorem gruop 8443
Description: A Grothendieck's universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
gruop  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  <. A ,  B >.  e.  U )

Proof of Theorem gruop
StepHypRef Expression
1 dfopg 3810 . . 3  |-  ( ( A  e.  U  /\  B  e.  U )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
213adant1 973 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
3 simp1 955 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U  e.  Univ )
4 grusn 8442 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  { A }  e.  U )
543adant3 975 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { A }  e.  U )
6 grupr 8435 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { A ,  B }  e.  U
)
7 grupr 8435 . . 3  |-  ( ( U  e.  Univ  /\  { A }  e.  U  /\  { A ,  B }  e.  U )  ->  { { A } ,  { A ,  B } }  e.  U
)
83, 5, 6, 7syl3anc 1182 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { { A } ,  { A ,  B } }  e.  U )
92, 8eqeltrd 2370 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  <. A ,  B >.  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   {csn 3653   {cpr 3654   <.cop 3656   Univcgru 8428
This theorem is referenced by:  gruf  8449
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-3an 936  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-tr 4130  df-iota 5235  df-fv 5279  df-ov 5877  df-gru 8429
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