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Theorem gruop 8427
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 3794 . . 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 8426 . . . 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 8419 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { A ,  B }  e.  U
)
7 grupr 8419 . . 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 2357 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 1623    e. wcel 1684   {csn 3640   {cpr 3641   <.cop 3643   Univcgru 8412
This theorem is referenced by:  gruf  8433
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-3an 936  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-tr 4114  df-iota 5219  df-fv 5263  df-ov 5861  df-gru 8413
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